Theorem disjex 28212

Description: Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)

Assertion

Ref	Expression
disjex	|- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( A = B \/ ( A i^i B ) = (/) ) )

Distinct variable groups:   z, A   z, B

Proof of Theorem disjex

Step	Hyp	Ref	Expression
1		orcom 393	. 2 |- ( ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
2		df-in 3379	. . . . . 6 |- ( A i^i B ) = { z | ( z e. A /\ z e. B ) }
3	2	neeq1i 2688	. . . . 5 |- ( ( A i^i B ) =/= (/) <-> { z | ( z e. A /\ z e. B ) } =/= (/) )
4		abn0 3719	. . . . 5 |- ( { z | ( z e. A /\ z e. B ) } =/= (/) <-> E. z ( z e. A /\ z e. B ) )
5	3, 4	bitr2i 258	. . . 4 |- ( E. z ( z e. A /\ z e. B ) <-> ( A i^i B ) =/= (/) )
6	5	necon2bbii 2675	. . 3 |- ( ( A i^i B ) = (/) <-> -. E. z ( z e. A /\ z e. B ) )
7	6	orbi2i 526	. 2 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) )
8		imor 418	. 2 |- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
9	1, 7, 8	3bitr4ri 286	1 |- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1448   E.wex 1667   e. wcel 1891   {cab 2438   =/= wne 2622   i^i cin 3371   (/)c0 3699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-v 3015  df-dif 3375  df-in 3379  df-nul 3700

This theorem is referenced by: (None)