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Theorem disjsn2 3670
Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
disjsn2 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)

Proof of Theorem disjsn2
StepHypRef Expression
1 elsni 3625 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2195 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 2409 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 3669 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 134 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2160  wne 2360  cin 3143  c0 3437  {csn 3607
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438  df-sn 3613
This theorem is referenced by:  disjpr2  3671  difprsn1  3746  diftpsn3  3748  xpsndisj  5073  funprg  5285  funtp  5288  f1oprg  5524  xp01disjl  6459  enpr2d  6843  phplem1  6880  prfidisj  6955  djuinr  7092  pm54.43  7219  pr2nelem  7220  sumpr  11453  setsfun0  12548  setscom  12552
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