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Theorem disjsn 3499
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 3330 . 2 ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}))
2 con2b 628 . . . 4 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥𝐴))
3 velsn 3458 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
43imbi1i 236 . . . 4 ((𝑥 ∈ {𝐵} → ¬ 𝑥𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥𝐴))
5 imnan 659 . . . 4 ((𝑥 = 𝐵 → ¬ 𝑥𝐴) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
62, 4, 53bitri 204 . . 3 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
76albii 1404 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴))
8 alnex 1433 . . 3 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
9 df-clel 2084 . . 3 (𝐵𝐴 ↔ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
108, 9xchbinxr 643 . 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 1287   = wceq 1289  wex 1426  wcel 1438  cin 2996  c0 3284  {csn 3441
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-in 3003  df-nul 3285  df-sn 3447
This theorem is referenced by:  disjsn2  3500  ssdifsn  3563  orddisj  4352  ndmima  4796  funtpg  5051  fnunsn  5107  ressnop0  5462  ftpg  5465  fsnunf  5480  fsnunfv  5481  phpm  6561  fiunsnnn  6577  ac6sfi  6594  unsnfi  6609  tpfidisj  6618  iunfidisj  6634  pm54.43  6797  fzpreddisj  9452  fzp1disj  9461  frecfzennn  9798  hashunsng  10180  hashxp  10199  fsumsplitsn  10767  sumtp  10771  fsumsplitsnun  10776  fsum2dlemstep  10791  fsumconst  10811  fsumabs  10822  fsumiun  10833
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