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Theorem dfdisj2 3775
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 3774 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
2 df-rmo 2331 . . 3  |-  ( E* x  e.  A  y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1375 . 2  |-  ( A. y E* x  e.  A  y  e.  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
41, 3bitri 177 1  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   A.wal 1257    e. wcel 1409   E*wmo 1917   E*wrmo 2326  Disj wdisj 3773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-rmo 2331  df-disj 3774
This theorem is referenced by:  disjss1  3779  nfdisjv  3785  invdisj  3787  sndisj  3788  disjxsn  3790
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