Theorem abssexg 4168

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 4166	. 2 |- ( A e. V -> ~P A e. _V )
2		df-pw 3568	. . . 4 |- ~P A = { x | x C_ A }
3	2	eleq1i 2236	. . 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 3219	. . . 4 |- { x | ( x C_ A /\ ph ) } C_ { x | x C_ A }
6		ssexg 4128	. . . 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 422	. . 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 2141   {cab 2156   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pmex  6631  tgval  12843