Theorem pwss 3668

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 3215	. 2 |- ( ~P A C_ B <-> A. x ( x e. ~P A -> x e. B ) )
2		df-pw 3654	. . . . 5 |- ~P A = { x | x C_ A }
3	2	abeq2i 2342	. . . 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 1518	. 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 1395   e. wcel 2202   C_ wss 3200   ~Pcpw 3652

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  axpweq  4261  setind2  4638