Theorem pwssb 4003

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 4002	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissb 3870	. 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 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: (None)