Theorem ssextss 4922

Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)

Assertion

Ref	Expression
ssextss	|- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssextss

Step	Hyp	Ref	Expression
1		sspwb 4917	. 2 |- ( A C_ B <-> ~P A C_ ~P B )
2		dfss2 3591	. 2 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		selpw 4165	. . . 4 |- ( x e. ~P A <-> x C_ A )
4		selpw 4165	. . . 4 |- ( x e. ~P B <-> x C_ B )
5	3, 4	imbi12i 340	. . 3 |- ( ( x e. ~P A -> x e. ~P B ) <-> ( x C_ A -> x C_ B ) )
6	5	albii 1747	. 2 |- ( A. x ( x e. ~P A -> x e. ~P B ) <-> A. x ( x C_ A -> x C_ B ) )
7	1, 2, 6	3bitri 286	1 |- ( A C_ B <-> A. x ( x C_ A -> x C_ 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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by:  ssext  4923  nssss  4924