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

Theorem disjsn2 3657
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 3612 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
21eqcomd 2183 . . 3  |-  ( B  e.  { A }  ->  A  =  B )
32necon3ai 2396 . 2  |-  ( A  =/=  B  ->  -.  B  e.  { A } )
4 disjsn 3656 . 2  |-  ( ( { A }  i^i  { B } )  =  (/) 
<->  -.  B  e.  { A } )
53, 4sylibr 134 1  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1353    e. wcel 2148    =/= wne 2347    i^i cin 3130   (/)c0 3424   {csn 3594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2741  df-dif 3133  df-in 3137  df-nul 3425  df-sn 3600
This theorem is referenced by:  disjpr2  3658  difprsn1  3733  diftpsn3  3735  xpsndisj  5057  funprg  5268  funtp  5271  f1oprg  5507  xp01disjl  6437  enpr2d  6819  phplem1  6854  prfidisj  6928  djuinr  7064  pm54.43  7191  pr2nelem  7192  sumpr  11423  setsfun0  12500  setscom  12504
  Copyright terms: Public domain W3C validator