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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
→
ι
be given.
Let x2 of type
ι
be given.
Assume H0:
x2
∈
lam
x0
x1
.
Apply lamE with
x0
,
x1
,
x2
,
x2
∈
V_
(
ordsucc
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x3 .
ordsucc
(
9d271..
(
x1
x3
)
)
)
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type
ι
be given.
Assume H1:
(
λ x4 .
and
(
x4
∈
x0
)
(
∃ x5 .
and
(
x5
∈
x1
x4
)
(
x2
=
setsum
x4
x5
)
)
)
x3
.
Apply H1 with
x2
∈
V_
(
ordsucc
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x4 .
ordsucc
(
9d271..
(
x1
x4
)
)
)
)
)
)
.
Assume H2:
x3
∈
x0
.
Assume H3:
∃ x4 .
and
(
x4
∈
x1
x3
)
(
x2
=
setsum
x3
x4
)
.
Apply H3 with
x2
∈
V_
(
ordsucc
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x4 .
ordsucc
(
9d271..
(
x1
x4
)
)
)
)
)
)
.
Let x4 of type
ι
be given.
Assume H4:
(
λ x5 .
and
(
x5
∈
x1
x3
)
(
x2
=
setsum
x3
x5
)
)
x4
.
Apply H4 with
x2
∈
V_
(
ordsucc
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x5 .
ordsucc
(
9d271..
(
x1
x5
)
)
)
)
)
)
.
Assume H5:
x4
∈
x1
x3
.
Assume H6:
x2
=
setsum
x3
x4
.
Apply unknownprop_283f8fc5ea1a3c99f01ef684040d36f3b3e77f532f79e28eeabcc4dccf9b7028 with
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x5 .
ordsucc
(
9d271..
(
x1
x5
)
)
)
)
,
x2
.
Apply H6 with
λ x5 x6 .
x6
⊆
V_
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x7 .
ordsucc
(
9d271..
(
x1
x7
)
)
)
)
)
.
Claim L7:
...
...
Claim L8:
...
...
Apply Subq_tra with
setsum
x3
x4
,
V_
(
ordsucc
(
binunion
(
9d271..
x3
)
(
9d271..
x4
)
)
)
,
V_
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x5 .
ordsucc
(
9d271..
(
x1
x5
)
)
)
)
)
leaving 2 subgoals.
The subproof is completed by applying unknownprop_777c77f22b29f6f8c2e6d97adb732f86ed83246887d8d67f81eae5d41628c7a2 with
x3
,
x4
.
Apply V_Subq_2 with
ordsucc
(
binunion
(
9d271..
x3
)
(
9d271..
x4
)
)
,
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x5 .
ordsucc
(
9d271..
(
x1
x5
)
)
)
)
.
Let x5 of type
ι
be given.
Assume H9:
x5
∈
ordsucc
(
binunion
(
9d271..
x3
)
(
9d271..
x4
)
)
.
Apply ordsuccE with
binunion
(
9d271..
x3
)
(
9d271..
x4
)
,
x5
,
x5
∈
V_
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x6 .
ordsucc
(
9d271..
(
x1
x6
)
)
)
)
)
leaving 3 subgoals.
The subproof is completed by applying H9.
Assume H10:
x5
∈
binunion
(
9d271..
x3
)
(
9d271..
x4
)
.
Apply binunionE with
9d271..
x3
,
9d271..
x4
,
x5
,
x5
∈
V_
(
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x6 .
ordsucc
(
9d271..
(
x1
x6
)
)
)
)
)
leaving 3 subgoals.
The subproof is completed by applying H10.
Assume H11:
x5
∈
9d271..
x3
.
Apply V_I with
x5
,
9d271..
x0
,
binunion
(
ordsucc
(
9d271..
x0
)
)
(
famunion
x0
(
λ x6 .
ordsucc
(
9d271..
(
x1
...
)
)
)
)
leaving 2 subgoals.
...
...
...
...
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