Theorem pwss 3575

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 3131	. 2 |- ( ~P A C_ B <-> A. x ( x e. ~P A -> x e. B ) )
2		df-pw 3561	. . . . 5 |- ~P A = { x | x C_ A }
3	2	abeq2i 2277	. . . 4 |- ( x e. ~P A <-> x C_ A )
4	3	imbi1i 237	. . 3 |- ( ( x e. ~P A -> x e. B ) <-> ( x C_ A -> x e. B ) )
5	4	albii 1458	. 2 |- ( A. x ( x e. ~P A -> x e. B ) <-> A. x ( x C_ A -> x e. B ) )
6	1, 5	bitri 183	1 |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   e. wcel 2136   C_ wss 3116   ~Pcpw 3559

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  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-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  axpweq  4150  setind2  4517