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Theorem disj2 3724
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 3374 . 2  |-  A  C_  _V
2 reldisj 3720 . 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 184    = wceq 1369   _Vcvv 2970    \ cdif 3323    i^i cin 3325    C_ wss 3326   (/)c0 3635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-dif 3329  df-in 3333  df-ss 3340  df-nul 3636
This theorem is referenced by:  ssindif0  3730  intirr  5214  setsres  14200  setscom  14202  f1omvdco3  15953  psgnunilem5  15998  opsrtoslem2  17564  clscon  19032  cldsubg  19679  imadifxp  25937
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