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Theorem disjsn2 3533
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 3492 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2105 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 2316 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 3532 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 133 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1299  wcel 1448  wne 2267  cin 3020  c0 3310  {csn 3474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-v 2643  df-dif 3023  df-in 3027  df-nul 3311  df-sn 3480
This theorem is referenced by:  disjpr2  3534  difprsn1  3606  diftpsn3  3608  xpsndisj  4901  funprg  5109  funtp  5112  f1oprg  5343  xp01disjl  6261  phplem1  6675  prfidisj  6744  djuinr  6863  pm54.43  6957  pr2nelem  6958  sumpr  11021  setsfun0  11777  setscom  11781
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