Theorem sspwuni 4060

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 2806	. . . 4 |- x e. _V
2	1	elpw 3662	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2539	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3217	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3928	. 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 2202   A.wral 2511   C_ wss 3201   ~Pcpw 3656   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-uni 3899

This theorem is referenced by:  pwssb  4061  elpwpw  4062  elpwuni  4065  rintm  4068  dftr4  4197  iotass  5311  tfrlemibfn  6537  tfr1onlembfn  6553  tfrcllembfn  6566  uniixp  6933  fipwssg  7221  unirnioo  10252  restid  13396  lssintclm  14463  topgele  14823  topontopn  14831  unitg  14856  epttop  14884  resttopon  14965  txuni2  15050  txdis  15071  unirnblps  15216  unirnbl  15217