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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
SNo
x0
.
Apply binunionI1 with
{x1 ∈
ordsucc
(
SNoLev
x0
)
|
(
λ x2 .
or
(
x2
∈
x0
)
(
x2
=
SNoLev
x0
)
)
x1
}
,
{
(
λ x2 .
SetAdjoin
x2
(
Sing
1
)
)
x1
|x1 ∈
ordsucc
(
SNoLev
x0
)
,
not
(
(
λ x2 .
or
(
x2
∈
x0
)
(
x2
=
SNoLev
x0
)
)
x1
)
}
,
SNoLev
x0
.
Apply SepI with
ordsucc
(
SNoLev
x0
)
,
λ x1 .
(
λ x2 .
or
(
x2
∈
x0
)
(
x2
=
SNoLev
x0
)
)
x1
,
SNoLev
x0
leaving 2 subgoals.
The subproof is completed by applying ordsuccI2 with
SNoLev
x0
.
Apply orIR with
SNoLev
x0
∈
x0
,
SNoLev
x0
=
SNoLev
x0
.
Let x1 of type
ι
→
ι
→
ο
be given.
Assume H1:
x1
(
SNoLev
x0
)
(
SNoLev
x0
)
.
The subproof is completed by applying H1.
■