Theorem pwss 3631

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		ssalel 3180	. 2 |- ( ~P A C_ B <-> A. x ( x e. ~P A -> x e. B ) )
2		df-pw 3617	. . . . 5 |- ~P A = { x | x C_ A }
3	2	abeq2i 2315	. . . 4 |- ( x e. ~P A <-> x C_ A )
4	3	imbi1i 238	. . 3 |- ( ( x e. ~P A -> x e. B ) <-> ( x C_ A -> x e. B ) )
5	4	albii 1492	. 2 |- ( A. x ( x e. ~P A -> x e. B ) <-> A. x ( x C_ A -> x e. B ) )
6	1, 5	bitri 184	1 |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370   e. wcel 2175   C_ wss 3165   ~Pcpw 3615

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-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178  df-pw 3617

This theorem is referenced by:  axpweq  4214  setind2  4587