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Theorem disjsn 3553
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 3381 . 2 ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}))
2 con2b 641 . . . 4 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥𝐴))
3 velsn 3512 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
43imbi1i 237 . . . 4 ((𝑥 ∈ {𝐵} → ¬ 𝑥𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥𝐴))
5 imnan 662 . . . 4 ((𝑥 = 𝐵 → ¬ 𝑥𝐴) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
62, 4, 53bitri 205 . . 3 ((𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵𝑥𝐴))
76albii 1429 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴))
8 alnex 1458 . . 3 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
9 df-clel 2111 . . 3 (𝐵𝐴 ↔ ∃𝑥(𝑥 = 𝐵𝑥𝐴))
108, 9xchbinxr 655 . 2 (∀𝑥 ¬ (𝑥 = 𝐵𝑥𝐴) ↔ ¬ 𝐵𝐴)
111, 7, 103bitri 205 1 ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  cin 3038  c0 3331  {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-nul 3332  df-sn 3501
This theorem is referenced by:  disjsn2  3554  ssdifsn  3619  orddisj  4429  ndmima  4884  funtpg  5142  fnunsn  5198  ressnop0  5567  ftpg  5570  fsnunf  5586  fsnunfv  5587  enpr2d  6677  phpm  6725  fiunsnnn  6741  ac6sfi  6758  unsnfi  6773  tpfidisj  6782  iunfidisj  6800  pm54.43  7012  dju1en  7033  fzpreddisj  9791  fzp1disj  9800  frecfzennn  10139  hashunsng  10493  hashxp  10512  fsumsplitsn  11119  sumtp  11123  fsumsplitsnun  11128  fsum2dlemstep  11143  fsumconst  11163  fsumabs  11174  fsumiun  11186  ennnfonelemhf1o  11821  structcnvcnv  11870  fsumcncntop  12620
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