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Theorem disjsn2 4715
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 4644 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2736 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 2963 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 4714 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 233 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2104  wne 2938  cin 3946  c0 4321  {csn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-v 3474  df-dif 3950  df-in 3954  df-nul 4322  df-sn 4628
This theorem is referenced by:  disjpr2  4716  disjtpsn  4718  difprsn1  4802  otsndisj  5518  xpsndisj  6161  funprg  6601  funtp  6604  funcnvpr  6609  f1oprg  6877  xp01disjl  8494  enpr2dOLD  9052  phplem1OLD  9219  djuin  9915  pm54.43  9998  pr2nelemOLD  10000  f1oun2prg  14872  s3sndisj  14918  sumpr  15698  cshwsdisj  17036  setsfun0  17109  setscom  17117  gsumpr  19864  dmdprdpr  19960  dprdpr  19961  ablfac1eulem  19983  cnfldfunALT  21157  cnfldfunALTOLD  21158  m2detleib  22353  dishaus  23106  dissnlocfin  23253  xpstopnlem1  23533  perfectlem2  26969  cosnopne  32183  prodpr  32299  esumpr  33362  esum2dlem  33388  prodfzo03  33913  gg-cnfldfunALT  35484  onint1  35637  bj-disjsn01  36136  lindsadd  36784  poimirlem26  36817  sumpair  44021  perfectALTVlem2  46688
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