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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
StepHypRef 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 ) }
32neeq1i 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 ) )
53, 4bitr2i 258 . . . 4  |-  ( E. z ( z  e.  A  /\  z  e.  B )  <->  ( A  i^i  B )  =/=  (/) )
65necon2bbii 2675 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  -.  E. z
( z  e.  A  /\  z  e.  B
) )
76orbi2i 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 ) )
91, 7, 83bitr4ri 286 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 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)
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