Theorem ssext 4242

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 4241	. . 3 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
2		ssextss 4241	. . 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 3185	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		albiim 1498	. 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 1362   = wceq 1364   C_ wss 3144

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616

This theorem is referenced by: (None)