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Theorem disjr 3464
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 3319 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2178 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
3 disj 3463 . 2  |-  ( ( B  i^i  A )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
42, 3bitri 183 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448    i^i cin 3120   (/)c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415
This theorem is referenced by: (None)
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