Theorem sspwuni 4002

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 2766	. . . 4 |- x e. _V
2	1	elpw 3612	. . 3 |- ( x e. ~P B <-> x C_ B )
3	2	ralbii 2503	. 2 |- ( A. x e. A x e. ~P B <-> A. x e. A x C_ B )
4		dfss3 3173	. 2 |- ( A C_ ~P B <-> A. x e. A x e. ~P B )
5		unissb 3870	. 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 2167   A.wral 2475   C_ wss 3157   ~Pcpw 3606   U.cuni 3840

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608  df-uni 3841

This theorem is referenced by:  pwssb  4003  elpwpw  4004  elpwuni  4007  rintm  4010  dftr4  4137  iotass  5237  tfrlemibfn  6395  tfr1onlembfn  6411  tfrcllembfn  6424  uniixp  6789  fipwssg  7054  unirnioo  10065  restid  12952  lssintclm  14016  topgele  14349  topontopn  14357  unitg  14382  epttop  14410  resttopon  14491  txuni2  14576  txdis  14597  unirnblps  14742  unirnbl  14743