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Theorem disjsn 3478
Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
disjsn ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)

Proof of Theorem disjsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 disj1 3315 . 2 ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}))
2 con2b 626 . . . 4 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥𝐴))
3 velsn 3439 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
43imbi1i 236 . . . 4 ((𝑥 ∈ {𝐵} → ¬ 𝑥𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥𝐴))
5 imnan 657 . . . 4 ((𝑥 = 𝐵 → ¬ 𝑥𝐴) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
62, 4, 53bitri 204 . . 3 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
76albii 1400 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴))
8 alnex 1429 . . 3 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
9 df-clel 2079 . . 3 (𝐵𝐴 ↔ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
108, 9xchbinxr 641 . 2 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ 𝐵𝐴)
111, 7, 103bitri 204 1 ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  cin 2983  c0 3269  {csn 3422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-dif 2986  df-in 2990  df-nul 3270  df-sn 3428
This theorem is referenced by:  disjsn2  3479  ssdifsn  3542  orddisj  4325  ndmima  4764  funtpg  5018  fnunsn  5074  ressnop0  5420  ftpg  5423  fsnunf  5438  fsnunfv  5439  phpm  6511  fiunsnnn  6527  ac6sfi  6544  unsnfi  6556  pm54.43  6721  fzpreddisj  9378  fzp1disj  9387  frecfzennn  9722  hashunsng  10050  hashxp  10069
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