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Theorem disjsn2 4640
Description: Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
disjsn2 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)

Proof of Theorem disjsn2
StepHypRef Expression
1 elsni 4574 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2824 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 3038 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 4639 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 235 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  wne 3013  cin 3932  c0 4288  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-v 3494  df-dif 3936  df-in 3940  df-nul 4289  df-sn 4558
This theorem is referenced by:  disjpr2  4641  disjtpsn  4643  difprsn1  4725  otsndisj  5400  xpsndisj  6013  funprg  6401  funtp  6404  funcnvpr  6409  f1oprg  6652  xp01disjl  8110  enpr2d  8585  phplem1  8684  djuin  9335  pm54.43  9417  pr2nelem  9418  f1oun2prg  14267  s3sndisj  14315  sumpr  15091  cshwsdisj  16420  setsfun0  16507  setscom  16515  gsumpr  19004  dmdprdpr  19100  dprdpr  19101  ablfac1eulem  19123  cnfldfun  20485  m2detleib  21168  dishaus  21918  dissnlocfin  22065  xpstopnlem1  22345  perfectlem2  25733  cosnopne  30356  prodpr  30469  esumpr  31224  esum2dlem  31250  prodfzo03  31773  onint1  33694  bj-disjsn01  34161  lindsadd  34766  poimirlem26  34799  sumpair  41169  perfectALTVlem2  43764
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