Theorem ssextss 4306

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 4302	. 2 |- ( A C_ B <-> ~P A C_ ~P B )
2		ssalel 3212	. 2 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		vex 2802	. . . . 5 |- x e. _V
4	3	elpw 3655	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3655	. . . 4 |- ( x e. ~P B <-> x C_ B )
6	4, 5	imbi12i 239	. . 3 |- ( ( x e. ~P A -> x e. ~P B ) <-> ( x C_ A -> x C_ B ) )
7	6	albii 1516	. 2 |- ( A. x ( x e. ~P A -> x e. ~P B ) <-> A. x ( x C_ A -> x C_ B ) )
8	1, 2, 7	3bitri 206	1 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   e. wcel 2200   C_ wss 3197   ~Pcpw 3649

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672

This theorem is referenced by:  ssext  4307  nssssr  4308