Theorem pwssb 4612

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 4611	. 2 |- ( A C_ ~P B <-> U. A C_ B )
2		unissb 4469	. 2 |- ( U. A C_ B <-> A. x e. A x C_ B )
3	1, 2	bitri 264	1 |- ( A C_ ~P B <-> A. x e. A x C_ B )

Syntax hints:   <-> wb 196   A.wral 2912   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  ustuni  22030  metustfbas  22362  dmvlsiga  30192  1stmbfm  30322  2ndmbfm  30323  dya2iocucvr  30346  gneispace  38432