Theorem ssext 4307

Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem ssext

Step	Hyp	Ref	Expression
1		ssextss 4306	. . 3 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
2		ssextss 4306	. . 3 |- ( B C_ A <-> A. x ( x C_ B -> x C_ A ) )
3	1, 2	anbi12i 460	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( A. x ( x C_ A -> x C_ B ) /\ A. x ( x C_ B -> x C_ A ) ) )
4		eqss 3239	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		albiim 1533	. 2 |- ( A. x ( x C_ A <-> x C_ B ) <-> ( A. x ( x C_ A -> x C_ B ) /\ A. x ( x C_ B -> x C_ A ) ) )
6	3, 4, 5	3bitr4i 212	1 |- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   C_ wss 3197

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: (None)