Theorem pwssb 3893

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 3892	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissb 3761	. 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 2414   C_ wss 3066   ~Pcpw 3505   U.cuni 3731

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-uni 3732

This theorem is referenced by: (None)