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Theorem disjexc 28251
Description: A variant of disjex 28250, 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 16 . 2  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  -> 
( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B ) )
3 orcom 393 . . 3  |-  ( ( A  =  B  \/  -.  E. z ( z  e.  A  /\  z  e.  B ) )  <->  ( -.  E. z ( z  e.  A  /\  z  e.  B )  \/  A  =  B ) )
4 df-in 3422 . . . . . . 7  |-  ( A  i^i  B )  =  { z  |  ( z  e.  A  /\  z  e.  B ) }
54neeq1i 2699 . . . . . 6  |-  ( ( A  i^i  B )  =/=  (/)  <->  { z  |  ( z  e.  A  /\  z  e.  B ) }  =/=  (/) )
6 abn0 3762 . . . . . 6  |-  ( { z  |  ( z  e.  A  /\  z  e.  B ) }  =/=  (/)  <->  E. z ( z  e.  A  /\  z  e.  B ) )
75, 6bitr2i 258 . . . . 5  |-  ( E. z ( z  e.  A  /\  z  e.  B )  <->  ( A  i^i  B )  =/=  (/) )
87necon2bbii 2686 . . . 4  |-  ( ( A  i^i  B )  =  (/)  <->  -.  E. z
( z  e.  A  /\  z  e.  B
) )
98orbi2i 526 . . 3  |-  ( ( A  =  B  \/  ( A  i^i  B )  =  (/) )  <->  ( A  =  B  \/  -.  E. z ( z  e.  A  /\  z  e.  B ) ) )
10 imor 418 . . 3  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B )  <->  ( -.  E. z ( z  e.  A  /\  z  e.  B )  \/  A  =  B ) )
113, 9, 103bitr4i 285 . 2  |-  ( ( A  =  B  \/  ( A  i^i  B )  =  (/) )  <->  ( E. z ( z  e.  A  /\  z  e.  B )  ->  A  =  B ) )
122, 11sylibr 217 1  |-  ( ( E. z ( z  e.  A  /\  z  e.  B )  ->  x  =  y )  -> 
( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447    =/= wne 2632    i^i cin 3414   (/)c0 3742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-nul 3743
This theorem is referenced by: (None)
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