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Theorem disjsn2 3829
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 3798 . . . 4  |-  ( B  e.  { A }  ->  B  =  A )
21eqcomd 2409 . . 3  |-  ( B  e.  { A }  ->  A  =  B )
32necon3ai 2607 . 2  |-  ( A  =/=  B  ->  -.  B  e.  { A } )
4 disjsn 3828 . 2  |-  ( ( { A }  i^i  { B } )  =  (/) 
<->  -.  B  e.  { A } )
53, 4sylibr 204 1  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2567    i^i cin 3279   (/)c0 3588   {csn 3774
This theorem is referenced by:  disjpr2  3830  difprsn1  3895  diftpsn3  3897  xpsndisj  5255  funprg  5459  funtp  5462  f1oprg  5677  phplem1  7245  pm54.43  7843  pr2nelem  7844  f1oun2prg  11819  setscom  13452  xpsc0  13740  xpsc1  13741  dmdprdpr  15562  dprdpr  15563  ablfac1eulem  15585  dishaus  17400  xpstopnlem1  17794  perfectlem2  20967  sumpr  24171  esumpr  24410  onint1  26103  sumpair  27573  otsndisj  27953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589  df-sn 3780
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