Theorem sspwuni 4092

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 2824	. . . 4 |- x e. _V
2	1	elpw 3691	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2556	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3236	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3960	. 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 2209   A.wral 2528   C_ wss 3220   ~Pcpw 3685   U.cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931

This theorem is referenced by:  pwssb  4093  elpwpw  4094  elpwuni  4097  rintm  4100  dftr4  4229  iotass  5350  tfrlemibfn  6589  tfr1onlembfn  6605  tfrcllembfn  6618  uniixp  6993  fipwssg  7303  unirnioo  10354  restid  13581  lssintclm  14693  topgele  15053  topontopn  15061  unitg  15086  epttop  15114  resttopon  15195  txuni2  15280  txdis  15301  unirnblps  15446  unirnbl  15447