Theorem dfon2lem2 31689

Description: Lemma for dfon2 31697. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

Ref	Expression
dfon2lem2	|- U. { x | ( x C_ A /\ ph /\ ps ) } C_ A

Distinct variable group:   x, A

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

Proof of Theorem dfon2lem2

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( x C_ A /\ ph /\ ps ) -> x C_ A )
2	1	ss2abi 3674	. . 3 |- { x | ( x C_ A /\ ph /\ ps ) } C_ { x | x C_ A }
3		df-pw 4160	. . 3 |- ~P A = { x | x C_ A }
4	2, 3	sseqtr4i 3638	. 2 |- { x | ( x C_ A /\ ph /\ ps ) } C_ ~P A
5		sspwuni 4611	. 2 |- ( { x | ( x C_ A /\ ph /\ ps ) } C_ ~P A <-> U. { x | ( x C_ A /\ ph /\ ps ) } C_ A )
6	4, 5	mpbi 220	1 |- U. { x | ( x C_ A /\ ph /\ ps ) } C_ A

Syntax hints:   /\ w3a 1037   {cab 2608   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

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

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-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  dfon2lem3  31690  dfon2lem7  31694