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Theorem disjsn 3669
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 3488 . 2 ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}))
2 con2b 670 . . . 4 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥𝐴))
3 velsn 3624 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
43imbi1i 238 . . . 4 ((𝑥 ∈ {𝐵} → ¬ 𝑥𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥𝐴))
5 imnan 691 . . . 4 ((𝑥 = 𝐵 → ¬ 𝑥𝐴) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
62, 4, 53bitri 206 . . 3 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
76albii 1481 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴))
8 alnex 1510 . . 3 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
9 df-clel 2185 . . 3 (𝐵𝐴 ↔ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
108, 9xchbinxr 684 . 2 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ 𝐵𝐴)
111, 7, 103bitri 206 1 ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  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-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438  df-sn 3613
This theorem is referenced by:  disjsn2  3670  ssdifsn  3735  orddisj  4563  ndmima  5023  funtpg  5286  fnunsn  5342  ressnop0  5718  ftpg  5721  fsnunf  5737  fsnunfv  5738  enpr2d  6843  phpm  6893  fiunsnnn  6909  ac6sfi  6926  unsnfi  6947  tpfidisj  6956  iunfidisj  6975  pm54.43  7219  dju1en  7242  fzpreddisj  10101  fzp1disj  10110  frecfzennn  10457  hashunsng  10819  hashxp  10838  fsumsplitsn  11450  sumtp  11454  fsumsplitsnun  11459  fsum2dlemstep  11474  fsumconst  11494  fsumabs  11505  fsumiun  11517  fprodm1  11638  fprodunsn  11644  fprod2dlemstep  11662  fprodsplitsn  11673  ennnfonelemhf1o  12464  structcnvcnv  12528  fsumcncntop  14516
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