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Theorem disjmoOLD 4089
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 4076 . 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 4088 . 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 1540    = wceq 1642    e. wcel 1710   E*wmo 2210   A.wral 2619    i^i cin 3227   (/)c0 3531  Disj wdisj 4074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rmo 2627  df-v 2866  df-dif 3231  df-in 3235  df-nul 3532  df-disj 4075
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