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Theorem disjmoOLD 4432
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 4419 . 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 4431 . 2  |-  (Disj  i  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
41, 3bitr3i 251 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   E*wmo 2276   A.wral 2814    i^i cin 3475   (/)c0 3785  Disj wdisj 4417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rmo 2822  df-v 3115  df-dif 3479  df-in 3483  df-nul 3786  df-disj 4418
This theorem is referenced by: (None)
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