Theorem abssexg 4161

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 4159	. 2 |- ( A e. V -> ~P A e. _V )
2		df-pw 3561	. . . 4 |- ~P A = { x | x C_ A }
3	2	eleq1i 2232	. . 3 |- ( ~P A e. _V <-> { x | x C_ A } e. _V )
4		simpl 108	. . . . 5 |- ( ( x C_ A /\ ph ) -> x C_ A )
5	4	ss2abi 3214	. . . 4 |- { x | ( x C_ A /\ ph ) } C_ { x | x C_ A }
6		ssexg 4121	. . . 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 421	. . 3 |- ( { x | x C_ A } e. _V -> { x | ( x C_ A /\ ph ) } e. _V )
8	3, 7	sylbi 120	. 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 103   e. wcel 2136   {cab 2151   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  pmex  6619  tgval  12689