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Theorem disjr 3661
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 3525 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2442 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
3 disj 3660 . 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 1652    e. wcel 1725   A.wral 2697    i^i cin 3311   (/)c0 3620
This theorem is referenced by:  zfreg2  7556  kqdisj  17756  iccntr  18844  stoweidlem57  27773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621
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