Theorem sspwuni 4097

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 3694	. . 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 3965	. 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 )

This proof depends on syntax axioms:   <-> wb 105   e. wcel 2209   A.wral 2528   C_ wss 3220   ~Pcpw 3688   U.cuni 3935

This proof depends on 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 proof 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 3690  df-uni 3936

This theorem is used by:  pwssb  4098  elpwpw  4099  elpwuni  4102  rintm  4105  dftr4  4236  iotass  5358  tfrlemibfn  6603  tfr1onlembfn  6619  tfrcllembfn  6632  uniixp  7007  fipwssg  7318  unirnioo  10392  restid  13663  lssintclm  14830  topgele  15255  topontopn  15263  unitg  15288  epttop  15316  resttopon  15397  txuni2  15482  txdis  15503  unirnblps  15648  unirnbl  15649