Theorem ssext 1865

Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets.

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem ssext

Step	Hyp	Ref	Expression
1		ssextss 1864	. . 3 |- (A (_ B <-> A.x(x (_ A -> x (_ B))
2		ssextss 1864	. . 3 |- (B (_ A <-> A.x(x (_ B -> x (_ A))
3	1, 2	anbi12i 369	. 2 |- ((A (_ B /\ B (_ A) <-> (A.x(x (_ A -> x (_ B) /\ A.x(x (_ B -> x (_ A)))
4		eqss 1516	. 2 |- (A = B <-> (A (_ B /\ B (_ A))
5		albi 785	. 2 |- (A.x(x (_ A <-> x (_ B) <-> (A.x(x (_ A -> x (_ B) /\ A.x(x (_ B -> x (_ A)))
6	3, 4, 5	3bitr4 158	1 |- (A = B <-> A.x(x (_ A <-> x (_ B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = wceq 1091   (_ wss 1487

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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