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Theorem disjmoOLD 4161
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 4148 . 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 4160 . 2  |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
41, 3bitr3i 243 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 177    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   E*wmo 2259   A.wral 2670    i^i cin 3283   (/)c0 3592  Disj wdisj 4146
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 2389
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-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rmo 2678  df-v 2922  df-dif 3287  df-in 3291  df-nul 3593  df-disj 4147
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