Theorem sspwuni 3760

Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
sspwuni	|- ( A C_ ~P B <-> U. A C_ B )

Proof of Theorem sspwuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . 4 |- x e. _V
2	1	elpw 3388	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2372	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 2989	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3631	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
6	3, 4, 5	3bitr4i 210	1 |- ( A C_ ~P B <-> U. A C_ B )

Syntax hints:   <-> wb 103   e. wcel 1433   A.wral 2348   C_ wss 2973   ~Pcpw 3382   U.cuni 3601

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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

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-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-uni 3602

This theorem is referenced by:  pwssb  3761  elpwuni  3762  rintm  3765  dftr4  3880  iotass  4904  tfrlemibfn  5965  unirnioo  8996