Theorem disjex 27678

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 387	. 2 |- ( ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
2		df-in 3478	. . . . . 6 |- ( A i^i B ) = { z | ( z e. A /\ z e. B ) }
3	2	neeq1i 2742	. . . . 5 |- ( ( A i^i B ) =/= (/) <-> { z | ( z e. A /\ z e. B ) } =/= (/) )
4		abn0 3813	. . . . 5 |- ( { z | ( z e. A /\ z e. B ) } =/= (/) <-> E. z ( z e. A /\ z e. B ) )
5	3, 4	bitr2i 250	. . . 4 |- ( E. z ( z e. A /\ z e. B ) <-> ( A i^i B ) =/= (/) )
6	5	necon2bbii 2724	. . 3 |- ( ( A i^i B ) = (/) <-> -. E. z ( z e. A /\ z e. B ) )
7	6	orbi2i 519	. 2 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) )
8		imor 412	. 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 278	1 |- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   {cab 2442   =/= wne 2652   i^i cin 3470   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-in 3478  df-nul 3794

This theorem is referenced by: (None)