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Metamath Proof Explorer

Theorem fin2i

Description: Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015)

Ref Expression
Assertion fin2i 
|- ( ( ( A e. Fin2 /\ B C_ ~P A ) /\ ( B =/= (/) /\ [C.] Or B ) ) -> U. B e. B )

Proof

Step Hyp Ref Expression
1 neeq1 
 |-  ( y = B -> ( y =/= (/) <-> B =/= (/) ) )
2 soeq2 
 |-  ( y = B -> ( [C.] Or y <-> [C.] Or B ) )
3 1 2 anbi12d 
 |-  ( y = B -> ( ( y =/= (/) /\ [C.] Or y ) <-> ( B =/= (/) /\ [C.] Or B ) ) )
4 unieq 
 |-  ( y = B -> U. y = U. B )
5 id 
 |-  ( y = B -> y = B )
6 4 5 eleq12d 
 |-  ( y = B -> ( U. y e. y <-> U. B e. B ) )
7 3 6 imbi12d 
 |-  ( y = B -> ( ( ( y =/= (/) /\ [C.] Or y ) -> U. y e. y ) <-> ( ( B =/= (/) /\ [C.] Or B ) -> U. B e. B ) ) )
8 isfin2 
 |-  ( A e. Fin2 -> ( A e. Fin2 <-> A. y e. ~P ~P A ( ( y =/= (/) /\ [C.] Or y ) -> U. y e. y ) ) )
9 8 ibi 
 |-  ( A e. Fin2 -> A. y e. ~P ~P A ( ( y =/= (/) /\ [C.] Or y ) -> U. y e. y ) )
10 9 adantr 
 |-  ( ( A e. Fin2 /\ B C_ ~P A ) -> A. y e. ~P ~P A ( ( y =/= (/) /\ [C.] Or y ) -> U. y e. y ) )
11 pwexg 
 |-  ( A e. Fin2 -> ~P A e. _V )
12 elpw2g 
 |-  ( ~P A e. _V -> ( B e. ~P ~P A <-> B C_ ~P A ) )
13 11 12 syl 
 |-  ( A e. Fin2 -> ( B e. ~P ~P A <-> B C_ ~P A ) )
14 13 biimpar 
 |-  ( ( A e. Fin2 /\ B C_ ~P A ) -> B e. ~P ~P A )
15 7 10 14 rspcdva 
 |-  ( ( A e. Fin2 /\ B C_ ~P A ) -> ( ( B =/= (/) /\ [C.] Or B ) -> U. B e. B ) )
16 15 imp 
 |-  ( ( ( A e. Fin2 /\ B C_ ~P A ) /\ ( B =/= (/) /\ [C.] Or B ) ) -> U. B e. B )