Theorem pwssb 3951

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 3950	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissb 3819	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
3	1, 2	bitri 183	1 |- ( A C_ ~P B <-> A. x e. A x C_ B )

Syntax hints:   <-> wb 104   A.wral 2444   C_ wss 3116   ~Pcpw 3559   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-uni 3790

This theorem is referenced by: (None)