Theorem pwss 4175

Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem pwss

Step	Hyp	Ref	Expression
1		dfss2 3591	. 2 |- ( ~P A C_ B <-> A. x ( x e. ~P A -> x e. B ) )
2		selpw 4165	. . . 4 |- ( x e. ~P A <-> x C_ A )
3	2	imbi1i 339	. . 3 |- ( ( x e. ~P A -> x e. B ) <-> ( x C_ A -> x e. B ) )
4	3	albii 1747	. 2 |- ( A. x ( x e. ~P A -> x e. B ) <-> A. x ( x C_ A -> x e. B ) )
5	1, 4	bitri 264	1 |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   C_ wss 3574   ~Pcpw 4158

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-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  axpweq  4842  setind2  8611  axgroth5  9646  grothpw  9648  axgroth6  9650