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Theorem disj2 3842
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 3484 . 2  |-  A  C_  _V
2 reldisj 3838 . 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 187    = wceq 1437   _Vcvv 3080    \ cdif 3433    i^i cin 3435    C_ wss 3436   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762
This theorem is referenced by:  ssindif0  3848  intirr  5237  setsres  15150  setscom  15152  f1omvdco3  17089  psgnunilem5  17134  opsrtoslem2  18707  clscon  20443  cldsubg  21123  uniinn0  28165  imadifxp  28214
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