Theorem sspwuni 4011

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 2774	. . . 4 |- x e. _V
2	1	elpw 3621	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2511	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3181	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3879	. 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 2175   A.wral 2483   C_ wss 3165   ~Pcpw 3615   U.cuni 3849

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-uni 3850

This theorem is referenced by:  pwssb  4012  elpwpw  4013  elpwuni  4016  rintm  4019  dftr4  4146  iotass  5248  tfrlemibfn  6413  tfr1onlembfn  6429  tfrcllembfn  6442  uniixp  6807  fipwssg  7080  unirnioo  10094  restid  13053  lssintclm  14117  topgele  14472  topontopn  14480  unitg  14505  epttop  14533  resttopon  14614  txuni2  14699  txdis  14720  unirnblps  14865  unirnbl  14866