Theorem ssextss 4205

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 4201	. 2 |- ( A C_ B <-> ~P A C_ ~P B )
2		dfss2 3136	. 2 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		vex 2733	. . . . 5 |- x e. _V
4	3	elpw 3572	. . . 4 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3572	. . . 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 1463	. 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 1346   e. wcel 2141   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by:  ssext  4206  nssssr  4207