Theorem abssexg 3955

Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
abssexg	|- ( A e. V -> { x | ( x C_ A /\ ph ) } e. _V )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem abssexg

Step	Hyp	Ref	Expression
1		pwexg 3954	. 2 |- ( A e. V -> ~P A e. _V )
2		df-pw 3384	. . . 4 |- ~P A = { x | x C_ A }
3	2	eleq1i 2144	. . 3 |- ( ~P A e. _V <-> { x | x C_ A } e. _V )
4		simpl 107	. . . . 5 |- ( ( x C_ A /\ ph ) -> x C_ A )
5	4	ss2abi 3066	. . . 4 |- { x | ( x C_ A /\ ph ) } C_ { x | x C_ A }
6		ssexg 3917	. . . 4 |- ( ( { x | ( x C_ A /\ ph ) } C_ { x | x C_ A } /\ { x | x C_ A } e. _V ) -> { x | ( x C_ A /\ ph ) } e. _V )
7	5, 6	mpan 414	. . 3 |- ( { x | x C_ A } e. _V -> { x | ( x C_ A /\ ph ) } e. _V )
8	3, 7	sylbi 119	. 2 |- ( ~P A e. _V -> { x | ( x C_ A /\ ph ) } e. _V )
9	1, 8	syl 14	1 |- ( A e. V -> { x | ( x C_ A /\ ph ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by: (None)