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Theorem dfdisj2 3995
Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
dfdisj2  |-  (Disj  x  e.  A B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem dfdisj2
StepHypRef Expression
1 df-disj 3994 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
2 df-rmo 2551 . . 3  |-  ( E* x  e.  A y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1553 . 2  |-  ( A. y E* x  e.  A
y  e.  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
41, 3bitri 240 1  |-  (Disj  x  e.  A B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684   E*wmo 2144   E*wrmo 2546  Disj wdisj 3993
This theorem is referenced by:  disjss1  3999  disjmoOLD  4008  disjiunOLD  4014  sndisj  4015  disjxsn  4017  disjss3  4022  fsumiunOLD  12281  hashiunOLD  12282  vitalilem3  18965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-rmo 2551  df-disj 3994
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