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Proofgold Asset
asset id
f4d4617ee83aa8d5cff1b342b008cb7cf2ff4fed9a7e0718458e83941e7f1e27
asset hash
ffcf72323521dc5ebe9155630513e449f539ddebf838de4d622b02e2c0a6d7fb
bday / block
36281
tx
d5d11..
preasset
doc published by
Pr4zB..
Param
4402e..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Param
cf2df..
:
ι
→
(
ι
→
ι
→
ο
) →
ο
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
setminus
setminus
:
ι
→
ι
→
ι
Param
Sing
Sing
:
ι
→
ι
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
8b6ad..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 .
∀ x5 : ο .
(
(
x1
=
x2
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x3
⟶
∀ x6 : ο .
x6
)
⟶
(
x1
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x2
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
(
x3
=
x4
⟶
∀ x6 : ο .
x6
)
⟶
not
(
x0
x1
x2
)
⟶
not
(
x0
x1
x3
)
⟶
not
(
x0
x2
x3
)
⟶
not
(
x0
x1
x4
)
⟶
not
(
x0
x2
x4
)
⟶
not
(
x0
x3
x4
)
⟶
x5
)
⟶
x5
Definition
62523..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 .
∀ x6 : ο .
(
8b6ad..
x0
x1
x2
x3
x4
⟶
(
x1
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
not
(
x0
x1
x5
)
⟶
not
(
x0
x2
x5
)
⟶
not
(
x0
x3
x5
)
⟶
x0
x4
x5
⟶
x6
)
⟶
x6
Definition
5e84d..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
62523..
x0
x1
x2
x3
x4
x5
⟶
(
x1
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x2
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x3
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x4
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x5
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x0
x1
x6
)
⟶
not
(
x0
x2
x6
)
⟶
x0
x3
x6
⟶
not
(
x0
x4
x6
)
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
7f677..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
5e84d..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
not
(
x0
x1
x7
)
⟶
x0
x2
x7
⟶
not
(
x0
x3
x7
)
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
372ed..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
7f677..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
x0
x1
x8
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
58366..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
5e84d..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
x0
x2
x7
⟶
not
(
x0
x3
x7
)
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
3b695..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
58366..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
x0
x2
x8
⟶
not
(
x0
x3
x8
)
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
fba9e..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
62523..
x0
x1
x2
x3
x4
x5
⟶
(
x1
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x2
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x3
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x4
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x5
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x0
x1
x6
)
⟶
x0
x2
x6
⟶
x0
x3
x6
⟶
not
(
x0
x4
x6
)
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
8c395..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
fba9e..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
x0
x3
x7
⟶
not
(
x0
x4
x7
)
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
c0878..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
8c395..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
x0
x3
x8
⟶
not
(
x0
x4
x8
)
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
c5756..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 .
∀ x6 : ο .
(
8b6ad..
x0
x1
x2
x3
x4
⟶
(
x1
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x2
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x3
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
(
x4
=
x5
⟶
∀ x7 : ο .
x7
)
⟶
not
(
x0
x1
x5
)
⟶
not
(
x0
x2
x5
)
⟶
x0
x3
x5
⟶
x0
x4
x5
⟶
x6
)
⟶
x6
Definition
2de86..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 .
∀ x7 : ο .
(
c5756..
x0
x1
x2
x3
x4
x5
⟶
(
x1
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x2
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x3
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x4
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
(
x5
=
x6
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x0
x1
x6
)
⟶
x0
x2
x6
⟶
not
(
x0
x3
x6
)
⟶
x0
x4
x6
⟶
not
(
x0
x5
x6
)
⟶
x7
)
⟶
x7
Definition
3109c..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 .
∀ x8 : ο .
(
2de86..
x0
x1
x2
x3
x4
x5
x6
⟶
(
x1
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x2
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x3
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x4
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x5
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
(
x6
=
x7
⟶
∀ x9 : ο .
x9
)
⟶
x0
x1
x7
⟶
not
(
x0
x2
x7
)
⟶
not
(
x0
x3
x7
)
⟶
x0
x4
x7
⟶
not
(
x0
x5
x7
)
⟶
not
(
x0
x6
x7
)
⟶
x8
)
⟶
x8
Definition
83424..
:=
λ x0 :
ι →
ι → ο
.
λ x1 x2 x3 x4 x5 x6 x7 x8 .
∀ x9 : ο .
(
3109c..
x0
x1
x2
x3
x4
x5
x6
x7
⟶
(
x1
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x2
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x3
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x4
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x5
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x6
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
(
x7
=
x8
⟶
∀ x10 : ο .
x10
)
⟶
not
(
x0
x1
x8
)
⟶
not
(
x0
x2
x8
)
⟶
not
(
x0
x3
x8
)
⟶
x0
x4
x8
⟶
not
(
x0
x5
x8
)
⟶
not
(
x0
x6
x8
)
⟶
not
(
x0
x7
x8
)
⟶
x9
)
⟶
x9
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
setminusE
setminusE
:
∀ x0 x1 x2 .
x2
∈
setminus
x0
x1
⟶
and
(
x2
∈
x0
)
(
nIn
x2
x1
)
Known
71109..
:
∀ x0 x1 .
∀ x2 :
ι →
ι → ο
.
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
x2
x3
x4
⟶
x2
x4
x3
)
⟶
4402e..
x1
x2
⟶
cf2df..
x1
x2
⟶
∀ x3 .
x3
∈
x1
⟶
x0
⊆
setminus
x1
(
Sing
x3
)
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
7f677..
x2
x4
x5
x6
x7
x8
x9
x10
⟶
∀ x11 : ο .
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
x2
x5
x3
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
not
(
x2
x10
x3
)
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
x2
x6
x3
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
not
(
x2
x9
x3
)
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
x2
x7
x3
⟶
not
(
x2
x8
x3
)
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
(
x2
x4
x3
⟶
not
(
x2
x5
x3
)
⟶
not
(
x2
x6
x3
)
⟶
not
(
x2
x7
x3
)
⟶
x2
x8
x3
⟶
x2
x9
x3
⟶
x2
x10
x3
⟶
x11
)
⟶
x11
Known
936d3..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
7f677..
x1
x2
x3
x4
x5
x6
x7
x8
⟶
7f677..
x1
x2
x8
x7
x5
x6
x4
x3
Known
neq_i_sym
neq_i_sym
:
∀ x0 x1 .
(
x0
=
x1
⟶
∀ x2 : ο .
x2
)
⟶
x1
=
x0
⟶
∀ x2 : ο .
x2
Known
e6ce7..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
62523..
x1
x2
x3
x4
x5
x6
⟶
62523..
x1
x2
x4
x3
x5
x6
Known
d257b..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
8b6ad..
x1
x2
x3
x4
x5
⟶
8b6ad..
x1
x3
x4
x2
x5
Known
da9f0..
:
∀ x0 .
∀ x1 :
ι →
ι → ο
.
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
x1
x2
x3
⟶
x1
x3
x2
)
⟶
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x0
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
8b6ad..
x1
x2
x3
x4
x5
⟶
8b6ad..
x1
x3
x2
x5
x4
Known
Subq_tra
Subq_tra
:
∀ x0 x1 x2 .
x0
⊆
x1
⟶
x1
⊆
x2
⟶
x0
⊆
x2
Known
setminus_Subq
setminus_Subq
:
∀ x0 x1 .
setminus
x0
x1
⊆
x0
Known
SingI
SingI
:
∀ x0 .
x0
∈
Sing
x0
Theorem
e1de9..
:
∀ x0 x1 .
∀ x2 :
ι →
ι → ο
.
(
∀ x3 .
x3
∈
x1
⟶
∀ x4 .
x4
∈
x1
⟶
x2
x3
x4
⟶
x2
x4
x3
)
⟶
4402e..
x1
x2
⟶
cf2df..
x1
x2
⟶
∀ x3 .
x3
∈
x1
⟶
x0
⊆
setminus
x1
(
Sing
x3
)
⟶
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x0
⟶
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x0
⟶
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x0
⟶
∀ x10 .
x10
∈
x0
⟶
7f677..
x2
x4
x5
x6
x7
x8
x9
x10
⟶
∀ x11 : ο .
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
372ed..
x2
x12
x13
x14
x15
x16
x17
x18
x3
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
3b695..
x2
x12
x13
x14
x15
x16
x17
x3
x18
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
c0878..
x2
x12
x13
x3
x14
x15
x16
x17
x18
⟶
x11
)
⟶
(
∀ x12 .
x12
∈
x0
⟶
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x0
⟶
∀ x15 .
x15
∈
x0
⟶
∀ x16 .
x16
∈
x0
⟶
∀ x17 .
x17
∈
x0
⟶
∀ x18 .
x18
∈
x0
⟶
83424..
x2
x12
x13
x14
x3
x15
x16
x17
x18
⟶
x11
)
⟶
x11
(proof)