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Theorem disjr 3820
Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
disjr  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
Distinct variable groups:    x, A    x, B

Proof of Theorem disjr
StepHypRef Expression
1 incom 3643 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2458 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
3 disj 3819 . 2  |-  ( ( B  i^i  A )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
42, 3bitri 249 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2795    i^i cin 3427   (/)c0 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-v 3072  df-dif 3431  df-in 3435  df-nul 3738
This theorem is referenced by:  zfreg2  7914  kqdisj  19423  iccntr  20516  stoweidlem57  29992
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