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Theorem disjexc 23986
Description: A variant of disjex 23985, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
Hypothesis
Ref Expression
disjexc.1  |-  ( x  =  y  ->  A  =  B )
Assertion
Ref Expression
disjexc  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  -> 
( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Distinct variable groups:    z, A    z, B
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem disjexc
StepHypRef Expression
1 disjexc.1 . . 3  |-  ( x  =  y  ->  A  =  B )
21imim2i 14 . 2  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  -> 
( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B ) )
3 orcom 377 . . 3  |-  ( ( A  =  B  \/  -.  E. z ( z  e.  A  /\  z  e.  B ) )  <->  ( -.  E. z ( z  e.  A  /\  z  e.  B )  \/  A  =  B ) )
4 df-in 3287 . . . . . . 7  |-  ( A  i^i  B )  =  { z  |  ( z  e.  A  /\  z  e.  B ) }
54neeq1i 2577 . . . . . 6  |-  ( ( A  i^i  B )  =/=  (/)  <->  { z  |  ( z  e.  A  /\  z  e.  B ) }  =/=  (/) )
6 abn0 3606 . . . . . 6  |-  ( { z  |  ( z  e.  A  /\  z  e.  B ) }  =/=  (/)  <->  E. z ( z  e.  A  /\  z  e.  B ) )
75, 6bitr2i 242 . . . . 5  |-  ( E. z ( z  e.  A  /\  z  e.  B )  <->  ( A  i^i  B )  =/=  (/) )
87necon2bbii 2623 . . . 4  |-  ( ( A  i^i  B )  =  (/)  <->  -.  E. z
( z  e.  A  /\  z  e.  B
) )
98orbi2i 506 . . 3  |-  ( ( A  =  B  \/  ( A  i^i  B )  =  (/) )  <->  ( A  =  B  \/  -.  E. z ( z  e.  A  /\  z  e.  B ) ) )
10 imor 402 . . 3  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B )  <->  ( -.  E. z ( z  e.  A  /\  z  e.  B )  \/  A  =  B ) )
113, 9, 103bitr4i 269 . 2  |-  ( ( A  =  B  \/  ( A  i^i  B )  =  (/) )  <->  ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B ) )
122, 11sylibr 204 1  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  -> 
( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567    i^i cin 3279   (/)c0 3588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589
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