Theorem disjex 29405

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 402	. 2 |- ( ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
2		df-in 3581	. . . . . 6 |- ( A i^i B ) = { z | ( z e. A /\ z e. B ) }
3	2	neeq1i 2858	. . . . 5 |- ( ( A i^i B ) =/= (/) <-> { z | ( z e. A /\ z e. B ) } =/= (/) )
4		abn0 3954	. . . . 5 |- ( { z | ( z e. A /\ z e. B ) } =/= (/) <-> E. z ( z e. A /\ z e. B ) )
5	3, 4	bitr2i 265	. . . 4 |- ( E. z ( z e. A /\ z e. B ) <-> ( A i^i B ) =/= (/) )
6	5	necon2bbii 2845	. . 3 |- ( ( A i^i B ) = (/) <-> -. E. z ( z e. A /\ z e. B ) )
7	6	orbi2i 541	. 2 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) )
8		imor 428	. 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 293	1 |- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by: (None)