Theorem pwssb 4061

Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Assertion

Ref	Expression
pwssb	|- ( A C_ ~P B <-> A. x e. A x C_ B )

Distinct variable groups:   x, A   x, B

Proof of Theorem pwssb

Step	Hyp	Ref	Expression
1		sspwuni 4060	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissb 3928	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
3	1, 2	bitri 184	1 |- ( A C_ ~P B <-> A. x e. A x C_ B )

Syntax hints:   <-> wb 105   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: (None)