Theorem fin23lem7 9138

Description: Lemma for isfin2-2 9141. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin23lem7	|- ( ( A e. V /\ B C_ ~P A /\ B =/= (/) ) -> { x e. ~P A | ( A \ x ) e. B } =/= (/) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem fin23lem7

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . . 4 |- ( B =/= (/) <-> E. y y e. B )
2		difss 3737	. . . . . . . 8 |- ( A \ y ) C_ A
3		elpw2g 4827	. . . . . . . . 9 |- ( A e. V -> ( ( A \ y ) e. ~P A <-> ( A \ y ) C_ A ) )
4	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> ( ( A \ y ) e. ~P A <-> ( A \ y ) C_ A ) )
5	2, 4	mpbiri 248	. . . . . . 7 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> ( A \ y ) e. ~P A )
6		simpr 477	. . . . . . . . . . 11 |- ( ( A e. V /\ B C_ ~P A ) -> B C_ ~P A )
7	6	sselda 3603	. . . . . . . . . 10 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> y e. ~P A )
8	7	elpwid 4170	. . . . . . . . 9 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> y C_ A )
9		dfss4 3858	. . . . . . . . 9 |- ( y C_ A <-> ( A \ ( A \ y ) ) = y )
10	8, 9	sylib 208	. . . . . . . 8 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> ( A \ ( A \ y ) ) = y )
11		simpr 477	. . . . . . . 8 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> y e. B )
12	10, 11	eqeltrd 2701	. . . . . . 7 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> ( A \ ( A \ y ) ) e. B )
13		difeq2 3722	. . . . . . . . 9 |- ( x = ( A \ y ) -> ( A \ x ) = ( A \ ( A \ y ) ) )
14	13	eleq1d 2686	. . . . . . . 8 |- ( x = ( A \ y ) -> ( ( A \ x ) e. B <-> ( A \ ( A \ y ) ) e. B ) )
15	14	rspcev 3309	. . . . . . 7 |- ( ( ( A \ y ) e. ~P A /\ ( A \ ( A \ y ) ) e. B ) -> E. x e. ~P A ( A \ x ) e. B )
16	5, 12, 15	syl2anc 693	. . . . . 6 |- ( ( ( A e. V /\ B C_ ~P A ) /\ y e. B ) -> E. x e. ~P A ( A \ x ) e. B )
17	16	ex 450	. . . . 5 |- ( ( A e. V /\ B C_ ~P A ) -> ( y e. B -> E. x e. ~P A ( A \ x ) e. B ) )
18	17	exlimdv 1861	. . . 4 |- ( ( A e. V /\ B C_ ~P A ) -> ( E. y y e. B -> E. x e. ~P A ( A \ x ) e. B ) )
19	1, 18	syl5bi 232	. . 3 |- ( ( A e. V /\ B C_ ~P A ) -> ( B =/= (/) -> E. x e. ~P A ( A \ x ) e. B ) )
20	19	3impia 1261	. 2 |- ( ( A e. V /\ B C_ ~P A /\ B =/= (/) ) -> E. x e. ~P A ( A \ x ) e. B )
21		rabn0 3958	. 2 |- ( { x e. ~P A | ( A \ x ) e. B } =/= (/) <-> E. x e. ~P A ( A \ x ) e. B )
22	20, 21	sylibr 224	1 |- ( ( A e. V /\ B C_ ~P A /\ B =/= (/) ) -> { x e. ~P A | ( A \ x ) e. B } =/= (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158

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

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-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160

This theorem is referenced by:  fin2i2  9140  isfin2-2  9141