Theorem ssextss 4221

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 4217	. 2 |- ( A C_ B <-> ~P A C_ ~P B )
2		dfss2 3145	. 2 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		vex 2741	. . . . 5 |- x e. _V
4	3	elpw 3582	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3582	. . . 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 1470	. 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 1351   e. wcel 2148   C_ wss 3130   ~Pcpw 3576

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599

This theorem is referenced by:  ssext  4222  nssssr  4223