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Theorem disjr 3496
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 3361 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2290 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
3 disj 3495 . 2  |-  ( ( B  i^i  A )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
42, 3bitri 240 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  zfreg2  7310  kqdisj  17423  iccntr  18326  bsstrs  26146  disjrOLD  26360  stoweidlem57  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456
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