ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  disj2 Unicode version

Theorem disj2 3418
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 3119 . 2  |-  A  C_  _V
2 reldisj 3414 . 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 set class
Syntax hints:    <-> wb 104    = wceq 1331   _Vcvv 2686    \ cdif 3068    i^i cin 3070    C_ wss 3071   (/)c0 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364
This theorem is referenced by:  ssindif0im  3422  intirr  4925  setsresg  12007  setscom  12009
  Copyright terms: Public domain W3C validator