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Theorem dfdisj2 4011
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 4010 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
2 df-rmo 2564 . . 3  |-  ( E* x  e.  A y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1556 . 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 1530    e. wcel 1696   E*wmo 2157   E*wrmo 2559  Disj wdisj 4009
This theorem is referenced by:  disjss1  4015  disjmoOLD  4024  disjiunOLD  4030  sndisj  4031  disjxsn  4033  disjss3  4038  fsumiunOLD  12297  hashiunOLD  12298  vitalilem3  18981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-rmo 2564  df-disj 4010
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