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Theorem dfdisj2 4419
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 4418 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
2 df-rmo 2822 . . 3  |-  ( E* x  e.  A  y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1620 . 2  |-  ( A. y E* x  e.  A  y  e.  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
41, 3bitri 249 1  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1377    e. wcel 1767   E*wmo 2276   E*wrmo 2817  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
This theorem depends on definitions:  df-bi 185  df-rmo 2822  df-disj 4418
This theorem is referenced by:  disjss1  4423  nfdisj  4429  disjmoOLD  4432  invdisj  4436  disjiunOLD  4438  sndisj  4439  disjxsn  4441  disjss3  4446  vitalilem3  21754
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