Theorem fin2i 9117

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

Assertion

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

Proof of Theorem fin2i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . . . 5 |- ( A e. FinII -> ~P A e. _V )
2		elpw2g 4827	. . . . 5 |- ( ~P A e. _V -> ( B e. ~P ~P A <-> B C_ ~P A ) )
3	1, 2	syl 17	. . . 4 |- ( A e. FinII -> ( B e. ~P ~P A <-> B C_ ~P A ) )
4	3	biimpar 502	. . 3 |- ( ( A e. FinII /\ B C_ ~P A ) -> B e. ~P ~P A )
5		isfin2 9116	. . . . 5 |- ( A e. FinII -> ( A e. FinII <-> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) ) )
6	5	ibi 256	. . . 4 |- ( A e. FinII -> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) )
7	6	adantr 481	. . 3 |- ( ( A e. FinII /\ B C_ ~P A ) -> A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) )
8		neeq1 2856	. . . . . 6 |- ( y = B -> ( y =/= (/) <-> B =/= (/) ) )
9		soeq2 5055	. . . . . 6 |- ( y = B -> ( [ C.] Or y <-> [ C.] Or B ) )
10	8, 9	anbi12d 747	. . . . 5 |- ( y = B -> ( ( y =/= (/) /\ [ C.] Or y ) <-> ( B =/= (/) /\ [ C.] Or B ) ) )
11		unieq 4444	. . . . . 6 |- ( y = B -> U. y = U. B )
12		id 22	. . . . . 6 |- ( y = B -> y = B )
13	11, 12	eleq12d 2695	. . . . 5 |- ( y = B -> ( U. y e. y <-> U. B e. B ) )
14	10, 13	imbi12d 334	. . . 4 |- ( y = B -> ( ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) <-> ( ( B =/= (/) /\ [ C.] Or B ) -> U. B e. B ) ) )
15	14	rspcv 3305	. . 3 |- ( B e. ~P ~P A -> ( A. y e. ~P ~P A ( ( y =/= (/) /\ [ C.] Or y ) -> U. y e. y ) -> ( ( B =/= (/) /\ [ C.] Or B ) -> U. B e. B ) ) )
16	4, 7, 15	sylc 65	. 2 |- ( ( A e. FinII /\ B C_ ~P A ) -> ( ( B =/= (/) /\ [ C.] Or B ) -> U. B e. B ) )
17	16	imp 445	1 |- ( ( ( A e. FinII /\ B C_ ~P A ) /\ ( B =/= (/) /\ [ C.] Or B ) ) -> U. B e. B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   Or wor 5034   [ C.] crpss 6936  Fin^IIcfin2 9101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-po 5035  df-so 5036  df-fin2 9108

This theorem is referenced by:  fin2i2  9140  ssfin2  9142  enfin2i  9143  fin1a2lem13  9234