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Theorem disjsn2 4608
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 4542 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2804 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 3012 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 4607 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 237 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2111  wne 2987  cin 3880  c0 4243  {csn 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-in 3888  df-nul 4244  df-sn 4526
This theorem is referenced by:  disjpr2  4609  disjtpsn  4611  difprsn1  4693  otsndisj  5374  xpsndisj  5987  funprg  6378  funtp  6381  funcnvpr  6386  f1oprg  6634  xp01disjl  8104  enpr2d  8580  phplem1  8680  djuin  9331  pm54.43  9414  pr2nelem  9415  f1oun2prg  14270  s3sndisj  14318  sumpr  15095  cshwsdisj  16424  setsfun0  16511  setscom  16519  gsumpr  19068  dmdprdpr  19164  dprdpr  19165  ablfac1eulem  19187  cnfldfun  20103  m2detleib  21236  dishaus  21987  dissnlocfin  22134  xpstopnlem1  22414  perfectlem2  25814  cosnopne  30454  prodpr  30568  esumpr  31435  esum2dlem  31461  prodfzo03  31984  onint1  33910  bj-disjsn01  34388  lindsadd  35050  poimirlem26  35083  sumpair  41662  perfectALTVlem2  44238
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