ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjsn2 Unicode version

Theorem disjsn2 3479
Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
disjsn2  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )

Proof of Theorem disjsn2
StepHypRef Expression
1 elsni 3440 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
21eqcomd 2088 . . 3  |-  ( B  e.  { A }  ->  A  =  B )
32necon3ai 2298 . 2  |-  ( A  =/=  B  ->  -.  B  e.  { A } )
4 disjsn 3478 . 2  |-  ( ( { A }  i^i  { B } )  =  (/) 
<->  -.  B  e.  { A } )
53, 4sylibr 132 1  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2249    i^i cin 2983   (/)c0 3269   {csn 3422
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2614  df-dif 2986  df-in 2990  df-nul 3270  df-sn 3428
This theorem is referenced by:  disjpr2  3480  difprsn1  3550  diftpsn3  3552  xpsndisj  4811  funprg  5017  funtp  5020  f1oprg  5243  phplem1  6498  prfidisj  6564  djuinr  6662  djuin  6663  pm54.43  6721  pr2nelem  6722
  Copyright terms: Public domain W3C validator