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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
StepHypRef 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 ) }
32neeq1i 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 ) )
53, 4bitr2i 250 . . . 4  |-  ( E. z ( z  e.  A  /\  z  e.  B )  <->  ( A  i^i  B )  =/=  (/) )
65necon2bbii 2724 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  -.  E. z
( z  e.  A  /\  z  e.  B
) )
76orbi2i 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 ) )
91, 7, 83bitr4ri 278 1  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B )  <->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Colors of variables: wff setvar class
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)
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