MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disj2 Structured version   Unicode version

Theorem disj2 3675
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 3368 . 2  |-  A  C_  _V
2 reldisj 3671 . 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 177    = wceq 1652   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  ssindif0  3681  intirr  5252  setsres  13495  setscom  13497  opsrtoslem2  16545  clscon  17493  cldsubg  18140  imadifxp  24038  f1omvdco3  27369  psgnunilem5  27394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
  Copyright terms: Public domain W3C validator