Theorem sspwuni 4049

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 2802	. . . 4 |- x e. _V
2	1	elpw 3655	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2536	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3213	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3917	. 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 2200   A.wral 2508   C_ wss 3197   ~Pcpw 3649   U.cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-uni 3888

This theorem is referenced by:  pwssb  4050  elpwpw  4051  elpwuni  4054  rintm  4057  dftr4  4186  iotass  5295  tfrlemibfn  6472  tfr1onlembfn  6488  tfrcllembfn  6501  uniixp  6866  fipwssg  7142  unirnioo  10165  restid  13278  lssintclm  14342  topgele  14697  topontopn  14705  unitg  14730  epttop  14758  resttopon  14839  txuni2  14924  txdis  14945  unirnblps  15090  unirnbl  15091