Theorem ssextss 4142

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 4138	. 2 |- ( A C_ B <-> ~P A C_ ~P B )
2		dfss2 3086	. 2 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		vex 2689	. . . . 5 |- x e. _V
4	3	elpw 3516	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3516	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	4, 5	imbi12i 238	. . 3 |- ( ( x e. ~P A -> x e. ~P B ) <-> ( x C_ A -> x C_ B ) )
7	6	albii 1446	. 2 |- ( A. x ( x e. ~P A -> x e. ~P B ) <-> A. x ( x C_ A -> x C_ B ) )
8	1, 2, 7	3bitri 205	1 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   C_ wss 3071   ~Pcpw 3510

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533

This theorem is referenced by:  ssext  4143  nssssr  4144