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Theorem disjmoOLD 4222
Description: Two ways to say that a collection  B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
disjmo.1  |-  ( i  =  j  ->  B  =  C )
Assertion
Ref Expression
disjmoOLD  |-  ( A. x E* i ( i  e.  A  /\  x  e.  B )  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Distinct variable groups:    i, j, x, A    B, j, x    C, i, x
Allowed substitution hints:    B( i)    C( j)

Proof of Theorem disjmoOLD
StepHypRef Expression
1 dfdisj2 4209 . 2  |-  (Disj  i  e.  A B  <->  A. x E* i ( i  e.  A  /\  x  e.  B ) )
2 disjmo.1 . . 3  |-  ( i  =  j  ->  B  =  C )
32disjor 4221 . 2  |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
41, 3bitr3i 244 1  |-  ( A. x E* i ( i  e.  A  /\  x  e.  B )  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1727   E*wmo 2288   A.wral 2711    i^i cin 3305   (/)c0 3613  Disj wdisj 4207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rmo 2719  df-v 2964  df-dif 3309  df-in 3313  df-nul 3614  df-disj 4208
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