ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  undisj2 Unicode version

Theorem undisj2 3303
Description: The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
undisj2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )

Proof of Theorem undisj2
StepHypRef Expression
1 un00 3291 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
2 indi 3212 . . 3  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
32eqeq1i 2089 . 2  |-  ( ( A  i^i  ( B  u.  C ) )  =  (/)  <->  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  =  (/) )
41, 3bitr4i 185 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285    u. cun 2972    i^i cin 2973   (/)c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator