Theorem sspwuni 4012

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 2775	. . . 4 |- x e. _V
2	1	elpw 3622	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2512	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3182	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3880	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
6	3, 4, 5	3bitr4i 212	1 |- ( A C_ ~P B <-> U. A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2176   A.wral 2484   C_ wss 3166   ~Pcpw 3616   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-uni 3851

This theorem is referenced by:  pwssb  4013  elpwpw  4014  elpwuni  4017  rintm  4020  dftr4  4147  iotass  5249  tfrlemibfn  6414  tfr1onlembfn  6430  tfrcllembfn  6443  uniixp  6808  fipwssg  7081  unirnioo  10095  restid  13082  lssintclm  14146  topgele  14501  topontopn  14509  unitg  14534  epttop  14562  resttopon  14643  txuni2  14728  txdis  14749  unirnblps  14894  unirnbl  14895