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Theorem disjr 3612
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 3476 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2394 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
3 disj 3611 . 2  |-  ( ( B  i^i  A )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
42, 3bitri 241 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2649    i^i cin 3262   (/)c0 3571
This theorem is referenced by:  zfreg2  7497  kqdisj  17685  iccntr  18723  stoweidlem57  27474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-dif 3266  df-in 3270  df-nul 3572
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