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Proofgold Proof
pf
Apply ZF_closed_E with
prim6
(
prim6
0
)
,
omega
∈
prim6
(
prim6
0
)
leaving 2 subgoals.
The subproof is completed by applying UnivOf_ZF_closed with
prim6
0
.
Assume H0:
Union_closed
(
prim6
(
prim6
0
)
)
.
Assume H1:
Power_closed
(
prim6
(
prim6
0
)
)
.
Assume H2:
Repl_closed
(
prim6
(
prim6
0
)
)
.
Claim L3:
omega
∈
prim4
(
prim6
0
)
Apply PowerI with
prim6
0
,
omega
.
Let x0 of type
ι
be given.
Assume H3:
x0
∈
omega
.
Apply nat_p_UnivOf_Empty with
x0
.
Apply omega_nat_p with
x0
.
The subproof is completed by applying H3.
Apply UnivOf_TransSet with
prim6
0
,
prim4
(
prim6
0
)
,
omega
leaving 2 subgoals.
Apply H1 with
prim6
0
.
The subproof is completed by applying UnivOf_In with
prim6
0
.
The subproof is completed by applying L3.
■