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Theorem disj2 3824
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
disj2  |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V 
\  B ) )

Proof of Theorem disj2
StepHypRef Expression
1 ssv 3464 . 2  |-  A  C_  _V
2 reldisj 3820 . 2  |-  ( A 
C_  _V  ->  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V 
\  B ) ) )
31, 2ax-mp 5 1  |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V 
\  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1455   _Vcvv 3057    \ cdif 3413    i^i cin 3415    C_ wss 3416   (/)c0 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744
This theorem is referenced by:  ssindif0  3830  intirr  5240  setsres  15206  setscom  15208  f1omvdco3  17145  psgnunilem5  17190  opsrtoslem2  18763  clscon  20500  cldsubg  21180  uniinn0  28218  imadifxp  28265
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