Theorem abssexg 4314

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 4312	. 2 |- ( A e. V -> ~P A e. _V )
2		df-pw 3687	. . . 4 |- ~P A = { x | x C_ A }
3	2	eleq1i 2304	. . 3 |- ( ~P A e. _V <-> { x | x C_ A } e. _V )
4		simpl 109	. . . . 5 |- ( ( x C_ A /\ ph ) -> x C_ A )
5	4	ss2abi 3320	. . . 4 |- { x | ( x C_ A /\ ph ) } C_ { x | x C_ A }
6		ssexg 4267	. . . 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 428	. . 3 |- ( { x | x C_ A } e. _V -> { x | ( x C_ A /\ ph ) } e. _V )
8	3, 7	sylbi 121	. 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 104   e. wcel 2209   {cab 2224   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  pmex  6917  tgval  13593