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Theorem disj2 3515
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 3211 . 2  |-  A  C_  _V
2 reldisj 3511 . 2  |-  ( A 
C_  _V  ->  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V 
\  B ) ) )
31, 2ax-mp 8 1  |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V 
\  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  ssindif0  3521  intirr  5077  setsres  13190  setscom  13192  opsrtoslem2  16242  clscon  17172  cldsubg  17809  f1omvdco3  27495  psgnunilem5  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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