Theorem ssfin2 9142

Description: A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
ssfin2	|- ( ( A e. FinII /\ B C_ A ) -> B e. FinII )

Proof of Theorem ssfin2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> A e. FinII )
2		elpwi 4168	. . . . . 6 |- ( x e. ~P ~P B -> x C_ ~P B )
3	2	adantl 482	. . . . 5 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> x C_ ~P B )
4		simplr 792	. . . . . 6 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> B C_ A )
5		sspwb 4917	. . . . . 6 |- ( B C_ A <-> ~P B C_ ~P A )
6	4, 5	sylib 208	. . . . 5 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> ~P B C_ ~P A )
7	3, 6	sstrd 3613	. . . 4 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> x C_ ~P A )
8		fin2i 9117	. . . . 5 |- ( ( ( A e. FinII /\ x C_ ~P A ) /\ ( x =/= (/) /\ [ C.] Or x ) ) -> U. x e. x )
9	8	ex 450	. . . 4 |- ( ( A e. FinII /\ x C_ ~P A ) -> ( ( x =/= (/) /\ [ C.] Or x ) -> U. x e. x ) )
10	1, 7, 9	syl2anc 693	. . 3 |- ( ( ( A e. FinII /\ B C_ A ) /\ x e. ~P ~P B ) -> ( ( x =/= (/) /\ [ C.] Or x ) -> U. x e. x ) )
11	10	ralrimiva 2966	. 2 |- ( ( A e. FinII /\ B C_ A ) -> A. x e. ~P ~P B ( ( x =/= (/) /\ [ C.] Or x ) -> U. x e. x ) )
12		ssexg 4804	. . . 4 |- ( ( B C_ A /\ A e. FinII ) -> B e. _V )
13	12	ancoms 469	. . 3 |- ( ( A e. FinII /\ B C_ A ) -> B e. _V )
14		isfin2 9116	. . 3 |- ( B e. _V -> ( B e. FinII <-> A. x e. ~P ~P B ( ( x =/= (/) /\ [ C.] Or x ) -> U. x e. x ) ) )
15	13, 14	syl 17	. 2 |- ( ( A e. FinII /\ B C_ A ) -> ( B e. FinII <-> A. x e. ~P ~P B ( ( x =/= (/) /\ [ C.] Or x ) -> U. x e. x ) ) )
16	11, 15	mpbird 247	1 |- ( ( A e. FinII /\ B C_ A ) -> B e. FinII )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-po 5035  df-so 5036  df-fin2 9108

This theorem is referenced by: (None)