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Proofgold Signed Transaction
vin
PrEvg..
/
1f042..
PUXi4..
/
44a66..
vout
PrEvg..
/
90f85..
0.32 bars
TMbij..
/
bc9ed..
negprop ownership controlledby
PrGxv..
upto 0
TMZFi..
/
a0836..
ownership of
48e77..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMZDd..
/
93b1b..
ownership of
04521..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMVFK..
/
aed29..
ownership of
fb6d0..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXYN..
/
e50d4..
ownership of
3dcb8..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMaFe..
/
fe1e9..
ownership of
96add..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMFfP..
/
738b6..
ownership of
2e5c3..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMaGz..
/
1085c..
ownership of
45a0f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKXs..
/
f96c6..
ownership of
7f087..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMGb8..
/
6630a..
ownership of
7fd4e..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRJT..
/
758ac..
ownership of
90122..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMVy1..
/
e252a..
ownership of
057f4..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMceb..
/
96c50..
ownership of
2e9d5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMR47..
/
8eaed..
ownership of
2cb0a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMMkx..
/
efb56..
ownership of
d7038..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMEo1..
/
380c4..
ownership of
3e6f6..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUV4..
/
dd18e..
ownership of
90bcc..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRdf..
/
b34ca..
ownership of
0cbec..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMFtU..
/
6cee3..
ownership of
9c135..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNHz..
/
6e968..
ownership of
8312b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdYq..
/
3f854..
ownership of
a89c5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMG13..
/
8fa5c..
ownership of
f1d4d..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMZq9..
/
7d560..
ownership of
bebd8..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMSxZ..
/
35c50..
ownership of
a87f5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdzu..
/
02497..
ownership of
6e6ce..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMWmq..
/
75723..
ownership of
5f7ef..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMV5a..
/
bd0eb..
ownership of
8e9af..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQ6w..
/
e6746..
ownership of
ac124..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMJ9u..
/
5f2fe..
ownership of
1fe1f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMY5a..
/
d7815..
ownership of
b5347..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQjX..
/
a6303..
ownership of
a538f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMGb4..
/
9f731..
ownership of
4a672..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMTq3..
/
ddde5..
ownership of
f8eb3..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPJW..
/
bcdb4..
ownership of
98ac3..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMMk4..
/
96774..
ownership of
b917a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMaRU..
/
c9a50..
ownership of
8ee89..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMHdA..
/
7f51a..
ownership of
f809b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXGW..
/
c48a5..
ownership of
a3321..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMVFn..
/
89325..
ownership of
b67f6..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQQd..
/
eddf9..
ownership of
c2711..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKxx..
/
bbd68..
ownership of
03f5f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMX6x..
/
a045e..
ownership of
6c085..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMGMC..
/
9a149..
ownership of
b57aa..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXTh..
/
0e321..
ownership of
4114a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMK56..
/
dce30..
ownership of
85e1b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMK6V..
/
c04c8..
ownership of
f3fb5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMTzL..
/
80a4f..
ownership of
67a9b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPFP..
/
f4711..
ownership of
b72a3..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKYU..
/
c7624..
ownership of
1f03c..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNGm..
/
331cb..
ownership of
a1a9f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMYDe..
/
746e8..
ownership of
d3e80..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMZ7J..
/
8e5d8..
ownership of
82e3b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMLFx..
/
547c6..
ownership of
45a16..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMYph..
/
6def6..
ownership of
8f913..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKvu..
/
1a5e4..
ownership of
43b91..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMWFu..
/
01f62..
ownership of
10a0b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMSb3..
/
9b936..
ownership of
e6223..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMMBf..
/
616b6..
ownership of
839f4..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMcFY..
/
4c427..
ownership of
f4ee1..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMJHY..
/
f55a6..
ownership of
be77e..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUHR..
/
ef2cf..
ownership of
aca6b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMM1n..
/
e8f2a..
ownership of
5c211..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKuo..
/
8086a..
ownership of
85336..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMZCB..
/
f3104..
ownership of
3238a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUMT..
/
4660d..
ownership of
ce785..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRkp..
/
67520..
ownership of
26358..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQ1e..
/
2442d..
ownership of
7e0c9..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMb58..
/
c52d4..
ownership of
d7fe6..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMK2y..
/
54b28..
ownership of
db15d..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRta..
/
d5af6..
ownership of
3db81..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUGa..
/
02b81..
ownership of
70a91..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQ6k..
/
2005d..
ownership of
2fbbc..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXft..
/
661fe..
ownership of
11171..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMd94..
/
7bac6..
ownership of
4c9b8..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdEf..
/
f244e..
ownership of
e5c15..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMW8a..
/
8dcd7..
ownership of
c1ec7..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMYAT..
/
b0cf6..
ownership of
a7e78..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMWQh..
/
e2b13..
ownership of
40f7a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQ2p..
/
559bd..
ownership of
d2696..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKsQ..
/
c40b0..
ownership of
64059..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMYKC..
/
d16d4..
ownership of
f4e77..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdck..
/
78ddd..
ownership of
5a21d..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdzY..
/
01804..
ownership of
0dbaa..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUSR..
/
64211..
ownership of
f336d..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXtp..
/
7cd50..
ownership of
94c43..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMU2x..
/
3264e..
ownership of
0b783..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMVnQ..
/
061e6..
ownership of
e256c..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPDx..
/
fd14b..
ownership of
f1083..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdQj..
/
335e4..
ownership of
375af..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQgn..
/
0900e..
ownership of
4b095..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMWg7..
/
8a66a..
ownership of
ce903..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMam4..
/
bb77c..
ownership of
054cd..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKz3..
/
84653..
ownership of
684cb..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNkU..
/
07abb..
ownership of
1b1d1..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMWLR..
/
7365a..
ownership of
2602c..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQUe..
/
e06e0..
ownership of
efbae..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMdhV..
/
93b22..
ownership of
e673f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQ3D..
/
67d9d..
ownership of
bba3d..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNLm..
/
0ebff..
ownership of
f1a52..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRtr..
/
b91d4..
ownership of
958ef..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPdr..
/
97786..
ownership of
dec29..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMMCB..
/
1bf8f..
ownership of
5faa3..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMGps..
/
22d81..
ownership of
38ce5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPNj..
/
5d46c..
ownership of
c79db..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNk4..
/
e18e6..
ownership of
4c4b2..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUCr..
/
18eb6..
ownership of
a5fe0..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMbMf..
/
e35c6..
ownership of
0353a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMHYF..
/
4f93c..
ownership of
695d8..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMGWw..
/
48431..
ownership of
badc5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXew..
/
50d25..
ownership of
3eff2..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMT98..
/
f6d11..
ownership of
1ac99..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUfi..
/
4cbe2..
ownership of
2d07f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMbWK..
/
7ecf8..
ownership of
0eb22..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMVhW..
/
e4d33..
ownership of
ac5c1..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMJw7..
/
d0c62..
ownership of
e75b5..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXxK..
/
01925..
ownership of
d0a1f..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMP3w..
/
24e7b..
ownership of
78dd4..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMXmt..
/
efe47..
ownership of
edd11..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPmQ..
/
a1933..
ownership of
b4672..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNcw..
/
b7664..
ownership of
287ca..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMQxi..
/
f1f5e..
ownership of
e4d6e..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMY5S..
/
0fa35..
ownership of
da962..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMPuf..
/
12345..
ownership of
0b5b6..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMKgz..
/
2df85..
ownership of
62c05..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMRrG..
/
7cfdf..
ownership of
1b39b..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMTPE..
/
38de8..
ownership of
c5d9a..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMNDB..
/
ede94..
ownership of
21ecc..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMMW9..
/
9e3c6..
ownership of
72294..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
PrGxv..
upto 0
TMUF1..
/
1fd79..
ownership of
0af09..
as prop with payaddr
PrGxv..
rightscost 0.00 controlledby
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upto 0
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f44cf..
doc published by
PrGxv..
Definition
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Theorem
and3I
:
∀ x0 x1 x2 : ο .
x0
⟶
x1
⟶
x2
⟶
and
(
and
x0
x1
)
x2
(proof)
Theorem
and3E
:
∀ x0 x1 x2 : ο .
and
(
and
x0
x1
)
x2
⟶
∀ x3 : ο .
(
x0
⟶
x1
⟶
x2
⟶
x3
)
⟶
x3
(proof)
Definition
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
orIL
:
∀ x0 x1 : ο .
x0
⟶
or
x0
x1
Theorem
or3I1
:
∀ x0 x1 x2 : ο .
x0
⟶
or
(
or
x0
x1
)
x2
(proof)
Known
orIR
:
∀ x0 x1 : ο .
x1
⟶
or
x0
x1
Theorem
or3I2
:
∀ x0 x1 x2 : ο .
x1
⟶
or
(
or
x0
x1
)
x2
(proof)
Theorem
or3I3
:
∀ x0 x1 x2 : ο .
x2
⟶
or
(
or
x0
x1
)
x2
(proof)
Theorem
or3E
:
∀ x0 x1 x2 : ο .
or
(
or
x0
x1
)
x2
⟶
∀ x3 : ο .
(
x0
⟶
x3
)
⟶
(
x1
⟶
x3
)
⟶
(
x2
⟶
x3
)
⟶
x3
(proof)
Theorem
and4I
:
∀ x0 x1 x2 x3 : ο .
x0
⟶
x1
⟶
x2
⟶
x3
⟶
and
(
and
(
and
x0
x1
)
x2
)
x3
(proof)
Theorem
and4E
:
∀ x0 x1 x2 x3 : ο .
and
(
and
(
and
x0
x1
)
x2
)
x3
⟶
∀ x4 : ο .
(
x0
⟶
x1
⟶
x2
⟶
x3
⟶
x4
)
⟶
x4
(proof)
Theorem
or4I1
:
∀ x0 x1 x2 x3 : ο .
x0
⟶
or
(
or
(
or
x0
x1
)
x2
)
x3
(proof)
Theorem
or4I2
:
∀ x0 x1 x2 x3 : ο .
x1
⟶
or
(
or
(
or
x0
x1
)
x2
)
x3
(proof)
Theorem
or4I3
:
∀ x0 x1 x2 x3 : ο .
x2
⟶
or
(
or
(
or
x0
x1
)
x2
)
x3
(proof)
Theorem
or4I4
:
∀ x0 x1 x2 x3 : ο .
x3
⟶
or
(
or
(
or
x0
x1
)
x2
)
x3
(proof)
Theorem
or4E
:
∀ x0 x1 x2 x3 : ο .
or
(
or
(
or
x0
x1
)
x2
)
x3
⟶
∀ x4 : ο .
(
x0
⟶
x4
)
⟶
(
x1
⟶
x4
)
⟶
(
x2
⟶
x4
)
⟶
(
x3
⟶
x4
)
⟶
x4
(proof)
Definition
Subq
:=
λ x0 x1 .
∀ x2 .
prim1
x2
x0
⟶
prim1
x2
x1
Definition
iff
:=
λ x0 x1 : ο .
and
(
x0
⟶
x1
)
(
x1
⟶
x0
)
Known
0ddd1..
:
∀ x0 x1 .
(
∀ x2 .
iff
(
prim1
x2
x0
)
(
prim1
x2
x1
)
)
⟶
x0
=
x1
Known
iffI
:
∀ x0 x1 : ο .
(
x0
⟶
x1
)
⟶
(
x1
⟶
x0
)
⟶
iff
x0
x1
Theorem
set_ext
:
∀ x0 x1 .
Subq
x0
x1
⟶
Subq
x1
x0
⟶
x0
=
x1
(proof)
Theorem
Subq_ref
:
∀ x0 .
Subq
x0
x0
(proof)
Theorem
Subq_tra
:
∀ x0 x1 x2 .
Subq
x0
x1
⟶
Subq
x1
x2
⟶
Subq
x0
x2
(proof)
Definition
False
:=
∀ x0 : ο .
x0
Definition
not
:=
λ x0 : ο .
x0
⟶
False
Known
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
FalseE
:
False
⟶
∀ x0 : ο .
x0
Theorem
dneg
:
∀ x0 : ο .
not
(
not
x0
)
⟶
x0
(proof)
Param
4a7ef..
:
ι
Definition
nIn
:=
λ x0 x1 .
not
(
prim1
x0
x1
)
Known
dcd83..
:
∀ x0 .
nIn
x0
4a7ef..
Theorem
2b26f..
:
not
(
∀ x0 : ο .
(
∀ x1 .
prim1
x1
4a7ef..
⟶
x0
)
⟶
x0
)
(proof)
Theorem
e8942..
:
∀ x0 .
Subq
4a7ef..
x0
(proof)
Param
c2e41..
:
ι
→
ι
→
ο
Known
adb47..
:
∀ x0 .
∀ x1 : ο .
(
∀ x2 .
and
(
and
(
and
(
prim1
x0
x2
)
(
∀ x3 x4 .
prim1
x3
x2
⟶
Subq
x4
x3
⟶
prim1
x4
x2
)
)
(
∀ x3 .
prim1
x3
x2
⟶
∀ x4 : ο .
(
∀ x5 .
and
(
prim1
x5
x2
)
(
∀ x6 .
Subq
x6
x3
⟶
prim1
x6
x5
)
⟶
x4
)
⟶
x4
)
)
(
∀ x3 .
Subq
x3
x2
⟶
or
(
c2e41..
x3
x2
)
(
prim1
x3
x2
)
)
⟶
x1
)
⟶
x1
Param
1216a..
:
ι
→
(
ι
→
ο
) →
ι
Known
492ff..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
∀ x3 : ο .
(
prim1
x2
x0
⟶
x1
x2
⟶
x3
)
⟶
x3
Known
b2421..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
x0
⟶
x1
x2
⟶
prim1
x2
(
1216a..
x0
x1
)
Theorem
b0a2d..
:
∀ x0 .
∀ x1 : ο .
(
∀ x2 .
(
∀ x3 .
iff
(
prim1
x3
x2
)
(
Subq
x3
x0
)
)
⟶
x1
)
⟶
x1
(proof)
Definition
e5b72..
:=
λ x0 .
prim0
(
λ x1 .
∀ x2 .
iff
(
prim1
x2
x1
)
(
Subq
x2
x0
)
)
Known
Eps_i_ex
:
∀ x0 :
ι → ο
.
(
∀ x1 : ο .
(
∀ x2 .
x0
x2
⟶
x1
)
⟶
x1
)
⟶
x0
(
prim0
x0
)
Theorem
cf34a..
:
∀ x0 x1 .
iff
(
prim1
x1
(
e5b72..
x0
)
)
(
Subq
x1
x0
)
(proof)
Theorem
b21da..
:
∀ x0 x1 .
prim1
x1
(
e5b72..
x0
)
⟶
Subq
x1
x0
(proof)
Theorem
3daee..
:
∀ x0 x1 .
Subq
x1
x0
⟶
prim1
x1
(
e5b72..
x0
)
(proof)
Theorem
72294..
:
∀ x0 x1 .
Subq
x0
x1
⟶
Subq
(
e5b72..
x0
)
(
e5b72..
x1
)
(proof)
Theorem
c5d9a..
:
∀ x0 .
prim1
4a7ef..
(
e5b72..
x0
)
(proof)
Theorem
62c05..
:
∀ x0 .
prim1
x0
(
e5b72..
x0
)
(proof)
Definition
0ac37..
:=
λ x0 x1 .
prim3
(
prim2
x0
x1
)
Known
UnionI
:
∀ x0 x1 x2 .
prim1
x1
x2
⟶
prim1
x2
x0
⟶
prim1
x1
(
prim3
x0
)
Known
67787..
:
∀ x0 x1 .
prim1
x0
(
prim2
x0
x1
)
Theorem
da962..
:
∀ x0 x1 x2 .
prim1
x2
x0
⟶
prim1
x2
(
0ac37..
x0
x1
)
(proof)
Known
5a932..
:
∀ x0 x1 .
prim1
x1
(
prim2
x0
x1
)
Theorem
287ca..
:
∀ x0 x1 x2 .
prim1
x2
x1
⟶
prim1
x2
(
0ac37..
x0
x1
)
(proof)
Known
UnionE_impred
:
∀ x0 x1 .
prim1
x1
(
prim3
x0
)
⟶
∀ x2 : ο .
(
∀ x3 .
prim1
x1
x3
⟶
prim1
x3
x0
⟶
x2
)
⟶
x2
Known
2532b..
:
∀ x0 x1 x2 .
prim1
x0
(
prim2
x1
x2
)
⟶
or
(
x0
=
x1
)
(
x0
=
x2
)
Theorem
edd11..
:
∀ x0 x1 x2 .
prim1
x2
(
0ac37..
x0
x1
)
⟶
or
(
prim1
x2
x0
)
(
prim1
x2
x1
)
(proof)
Known
6982e..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
and
(
prim1
x2
x0
)
(
x1
x2
)
Theorem
d0a1f..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
prim1
x2
x0
(proof)
Theorem
ac5c1..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
x1
x2
(proof)
Definition
d3786..
:=
λ x0 x1 .
1216a..
x0
(
λ x2 .
prim1
x2
x1
)
Theorem
2d07f..
:
∀ x0 x1 x2 .
prim1
x2
x0
⟶
prim1
x2
x1
⟶
prim1
x2
(
d3786..
x0
x1
)
(proof)
Theorem
3eff2..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
and
(
prim1
x2
x0
)
(
prim1
x2
x1
)
(proof)
Theorem
695d8..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
prim1
x2
x0
(proof)
Theorem
a5fe0..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
prim1
x2
x1
(proof)
Param
91630..
:
ι
→
ι
Definition
4ae4a..
:=
λ x0 .
0ac37..
x0
(
91630..
x0
)
Theorem
c79db..
:
∀ x0 .
Subq
x0
(
4ae4a..
x0
)
(proof)
Known
e7a4c..
:
∀ x0 .
prim1
x0
(
91630..
x0
)
Theorem
5faa3..
:
∀ x0 .
prim1
x0
(
4ae4a..
x0
)
(proof)
Known
fead7..
:
∀ x0 x1 .
prim1
x1
(
91630..
x0
)
⟶
x1
=
x0
Theorem
958ef..
:
∀ x0 x1 .
prim1
x1
(
4ae4a..
x0
)
⟶
or
(
prim1
x1
x0
)
(
x1
=
x0
)
(proof)
Known
113d7..
:
∀ x0 x1 .
prim1
x1
x0
⟶
∀ x2 : ο .
(
∀ x3 .
and
(
prim1
x3
x0
)
(
not
(
∀ x4 : ο .
(
∀ x5 .
and
(
prim1
x5
x0
)
(
prim1
x5
x3
)
⟶
x4
)
⟶
x4
)
)
⟶
x2
)
⟶
x2
Theorem
bba3d..
:
∀ x0 x1 .
prim1
x0
x1
⟶
nIn
x1
x0
(proof)
Theorem
efbae..
:
∀ x0 .
4a7ef..
=
4ae4a..
x0
⟶
∀ x1 : ο .
x1
(proof)
Theorem
1b1d1..
:
∀ x0 .
4ae4a..
x0
=
4a7ef..
⟶
∀ x1 : ο .
x1
(proof)
Theorem
054cd..
:
∀ x0 x1 .
4ae4a..
x0
=
4ae4a..
x1
⟶
x0
=
x1
(proof)
Theorem
4b095..
:
∀ x0 x1 .
(
x0
=
x1
⟶
∀ x2 : ο .
x2
)
⟶
4ae4a..
x0
=
4ae4a..
x1
⟶
∀ x2 : ο .
x2
(proof)
Theorem
f1083..
:
prim1
4a7ef..
(
4ae4a..
4a7ef..
)
(proof)
Theorem
0b783..
:
prim1
(
4ae4a..
4a7ef..
)
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(proof)
Theorem
f336d..
:
prim1
4a7ef..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(proof)
Definition
TransSet
:=
λ x0 .
∀ x1 .
prim1
x1
x0
⟶
Subq
x1
x0
Definition
ordinal
:=
λ x0 .
and
(
TransSet
x0
)
(
∀ x1 .
prim1
x1
x0
⟶
TransSet
x1
)
Known
andEL
:
∀ x0 x1 : ο .
and
x0
x1
⟶
x0
Theorem
ordinal_TransSet
:
∀ x0 .
ordinal
x0
⟶
TransSet
x0
(proof)
Known
andER
:
∀ x0 x1 : ο .
and
x0
x1
⟶
x1
Theorem
ordinal_In_TransSet
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
prim1
x1
x0
⟶
TransSet
x1
(proof)
Theorem
40f7a..
:
ordinal
4a7ef..
(proof)
Theorem
ordinal_Hered
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
prim1
x1
x0
⟶
ordinal
x1
(proof)
Definition
ba9d8..
:=
λ x0 .
∀ x1 :
ι → ο
.
x1
4a7ef..
⟶
(
∀ x2 .
x1
x2
⟶
x1
(
4ae4a..
x2
)
)
⟶
x1
x0
Theorem
4c9b8..
:
ba9d8..
4a7ef..
(proof)
Theorem
2fbbc..
:
∀ x0 .
ba9d8..
x0
⟶
ba9d8..
(
4ae4a..
x0
)
(proof)
Theorem
3db81..
:
ba9d8..
(
4ae4a..
4a7ef..
)
(proof)
Theorem
d7fe6..
:
ba9d8..
(
4ae4a..
(
4ae4a..
4a7ef..
)
)
(proof)
Theorem
26358..
:
∀ x0 .
ba9d8..
x0
⟶
prim1
4a7ef..
(
4ae4a..
x0
)
(proof)
Theorem
3238a..
:
∀ x0 .
ba9d8..
x0
⟶
∀ x1 .
prim1
x1
x0
⟶
prim1
(
4ae4a..
x1
)
(
4ae4a..
x0
)
(proof)
Theorem
5c211..
:
∀ x0 :
ι → ο
.
x0
4a7ef..
⟶
(
∀ x1 .
ba9d8..
x1
⟶
x0
x1
⟶
x0
(
4ae4a..
x1
)
)
⟶
∀ x1 .
ba9d8..
x1
⟶
x0
x1
(proof)
Theorem
be77e..
:
∀ x0 .
ba9d8..
x0
⟶
or
(
x0
=
4a7ef..
)
(
∀ x1 : ο .
(
∀ x2 .
and
(
ba9d8..
x2
)
(
x0
=
4ae4a..
x2
)
⟶
x1
)
⟶
x1
)
(proof)
Theorem
839f4..
:
∀ x0 :
ι → ο
.
(
∀ x1 .
ba9d8..
x1
⟶
(
∀ x2 .
prim1
x2
x1
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
ba9d8..
x1
⟶
x0
x1
(proof)
Theorem
10a0b..
:
∀ x0 .
ba9d8..
x0
⟶
∀ x1 .
prim1
x1
x0
⟶
ba9d8..
x1
(proof)
Theorem
8f913..
:
∀ x0 .
ba9d8..
x0
⟶
∀ x1 .
prim1
x1
x0
⟶
Subq
x1
x0
(proof)
Theorem
82e3b..
:
∀ x0 .
ba9d8..
x0
⟶
∀ x1 .
prim1
x1
(
4ae4a..
x0
)
⟶
Subq
x1
x0
(proof)
Theorem
a1a9f..
:
∀ x0 .
TransSet
x0
⟶
TransSet
(
4ae4a..
x0
)
(proof)
Theorem
b72a3..
:
∀ x0 .
ordinal
x0
⟶
ordinal
(
4ae4a..
x0
)
(proof)
Theorem
f3fb5..
:
∀ x0 .
ba9d8..
x0
⟶
ordinal
x0
(proof)
Theorem
4114a..
:
∀ x0 : ο .
(
∀ x1 .
(
∀ x2 .
iff
(
prim1
x2
x1
)
(
ba9d8..
x2
)
)
⟶
x0
)
⟶
x0
(proof)
Definition
48ef8..
:=
prim0
(
λ x0 .
∀ x1 .
iff
(
prim1
x1
x0
)
(
ba9d8..
x1
)
)
Theorem
6c085..
:
∀ x0 .
iff
(
prim1
x0
48ef8..
)
(
ba9d8..
x0
)
(proof)
Theorem
c2711..
:
∀ x0 .
prim1
x0
48ef8..
⟶
ba9d8..
x0
(proof)
Theorem
a3321..
:
∀ x0 .
ba9d8..
x0
⟶
prim1
x0
48ef8..
(proof)
Theorem
8ee89..
:
prim1
4a7ef..
48ef8..
(proof)
Theorem
98ac3..
:
∀ x0 .
prim1
x0
48ef8..
⟶
prim1
(
4ae4a..
x0
)
48ef8..
(proof)
Definition
aa8d2..
:=
λ x0 x1 x2 .
∀ x3 :
ι →
ι → ο
.
x3
4a7ef..
x1
⟶
(
∀ x4 .
ba9d8..
x4
⟶
∀ x5 .
x3
x4
x5
⟶
x3
(
4ae4a..
x4
)
(
prim3
x5
)
)
⟶
x3
x0
x2
Theorem
4a672..
:
∀ x0 .
aa8d2..
4a7ef..
x0
x0
(proof)
Theorem
b5347..
:
∀ x0 x1 .
ba9d8..
x1
⟶
∀ x2 .
aa8d2..
x1
x0
x2
⟶
aa8d2..
(
4ae4a..
x1
)
x0
(
prim3
x2
)
(proof)
Theorem
ac124..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
x1
4a7ef..
x0
⟶
(
∀ x2 .
ba9d8..
x2
⟶
∀ x3 .
aa8d2..
x2
x0
x3
⟶
x1
x2
x3
⟶
x1
(
4ae4a..
x2
)
(
prim3
x3
)
)
⟶
∀ x2 x3 .
ba9d8..
x2
⟶
aa8d2..
x2
x0
x3
⟶
x1
x2
x3
(proof)
Theorem
5f7ef..
:
∀ x0 x1 x2 .
ba9d8..
x1
⟶
aa8d2..
x1
x0
x2
⟶
or
(
and
(
x1
=
4a7ef..
)
(
x2
=
x0
)
)
(
∀ x3 : ο .
(
∀ x4 .
(
∀ x5 : ο .
(
∀ x6 .
and
(
and
(
x1
=
4ae4a..
x4
)
(
x2
=
prim3
x6
)
)
(
aa8d2..
x4
x0
x6
)
⟶
x5
)
⟶
x5
)
⟶
x3
)
⟶
x3
)
(proof)
Theorem
a87f5..
:
∀ x0 x1 .
aa8d2..
4a7ef..
x0
x1
⟶
x1
=
x0
(proof)
Theorem
f1d4d..
:
∀ x0 x1 x2 .
ba9d8..
x0
⟶
aa8d2..
(
4ae4a..
x0
)
x1
x2
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
x2
=
prim3
x4
)
(
aa8d2..
x0
x1
x4
)
⟶
x3
)
⟶
x3
(proof)
Theorem
8312b..
:
∀ x0 x1 .
ba9d8..
x1
⟶
∀ x2 : ο .
(
∀ x3 .
aa8d2..
x1
x0
x3
⟶
x2
)
⟶
x2
(proof)
Theorem
0cbec..
:
∀ x0 x1 .
ba9d8..
x1
⟶
∀ x2 x3 .
aa8d2..
x1
x0
x2
⟶
aa8d2..
x1
x0
x3
⟶
x2
=
x3
(proof)
Theorem
3e6f6..
:
∀ x0 x1 .
ba9d8..
x0
⟶
aa8d2..
x0
x1
(
prim0
(
aa8d2..
x0
x1
)
)
(proof)
Theorem
2cb0a..
:
∀ x0 .
prim0
(
aa8d2..
4a7ef..
x0
)
=
x0
(proof)
Theorem
057f4..
:
∀ x0 x1 .
ba9d8..
x0
⟶
prim0
(
aa8d2..
(
4ae4a..
x0
)
x1
)
=
prim3
(
prim0
(
aa8d2..
x0
x1
)
)
(proof)
Param
94f9e..
:
ι
→
(
ι
→
ι
) →
ι
Definition
2f2ea..
:=
λ x0 .
prim3
(
94f9e..
48ef8..
(
λ x1 .
prim0
(
aa8d2..
x1
x0
)
)
)
Known
696c0..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
(
94f9e..
x0
x1
)
Theorem
7fd4e..
:
∀ x0 x1 x2 .
ba9d8..
x2
⟶
prim1
x1
(
prim0
(
aa8d2..
x2
x0
)
)
⟶
prim1
x1
(
2f2ea..
x0
)
(proof)
Known
8c6f6..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
(
94f9e..
x0
x1
)
⟶
∀ x3 : ο .
(
∀ x4 .
prim1
x4
x0
⟶
x2
=
x1
x4
⟶
x3
)
⟶
x3
Theorem
45a0f..
:
∀ x0 x1 .
prim1
x1
(
2f2ea..
x0
)
⟶
∀ x2 : ο .
(
∀ x3 .
ba9d8..
x3
⟶
prim1
x1
(
prim0
(
aa8d2..
x3
x0
)
)
⟶
x2
)
⟶
x2
(proof)
Theorem
96add..
:
∀ x0 .
Subq
x0
(
2f2ea..
x0
)
(proof)
Theorem
fb6d0..
:
∀ x0 .
TransSet
(
2f2ea..
x0
)
(proof)
Theorem
In_ind
:
∀ x0 :
ι → ο
.
(
∀ x1 .
(
∀ x2 .
prim1
x2
x1
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
x0
x1
(proof)