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Theorem disjmoOLD 4008
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.)
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 3995 . 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 4007 . 2  |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
41, 3bitr3i 242 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 176    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144   A.wral 2543    i^i cin 3151   (/)c0 3455  Disj wdisj 3993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rmo 2551  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456  df-disj 3994
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