Theorem pwss 3374

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 2934	. 2 |- ( ~P A C_ B <-> A. x ( x e. ~P A -> x e. B ) )
2		df-pw 3361	. . . . 5 |- ~P A = { x | x C_ A }
3	2	abeq2i 2148	. . . 4 |- ( x e. ~P A <-> x C_ A )
4	3	imbi1i 227	. . 3 |- ( ( x e. ~P A -> x e. B ) <-> ( x C_ A -> x e. B ) )
5	4	albii 1359	. 2 |- ( A. x ( x e. ~P A -> x e. B ) <-> A. x ( x C_ A -> x e. B ) )
6	1, 5	bitri 173	1 |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   e. wcel 1393   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  axpweq  3924  setind2  4265