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Theorem disjsn2 3461
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 3421 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
21eqcomd 2061 . . 3 (𝐵 ∈ {𝐴} → 𝐴 = 𝐵)
32necon3ai 2269 . 2 (𝐴𝐵 → ¬ 𝐵 ∈ {𝐴})
4 disjsn 3460 . 2 (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴})
53, 4sylibr 141 1 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1259  wcel 1409  wne 2220  cin 2944  c0 3252  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2948  df-in 2952  df-nul 3253  df-sn 3409
This theorem is referenced by:  disjpr2  3462  difprsn1  3531  diftpsn3  3533  xpsndisj  4777  funprg  4977  funtp  4980  f1oprg  5196  phplem1  6346
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