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Theorem disjsn 3423
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}) = ∅ ↔ ¬ B A)

Proof of Theorem disjsn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 disj1 3264 . 2 ((A ∩ {B}) = ∅ ↔ x(x A → ¬ x {B}))
2 con2b 592 . . . 4 ((x A → ¬ x {B}) ↔ (x {B} → ¬ x A))
3 elsn 3382 . . . . 5 (x {B} ↔ x = B)
43imbi1i 227 . . . 4 ((x {B} → ¬ x A) ↔ (x = B → ¬ x A))
5 imnan 623 . . . 4 ((x = B → ¬ x A) ↔ ¬ (x = B x A))
62, 4, 53bitri 195 . . 3 ((x A → ¬ x {B}) ↔ ¬ (x = B x A))
76albii 1356 . 2 (x(x A → ¬ x {B}) ↔ x ¬ (x = B x A))
8 alnex 1385 . . 3 (x ¬ (x = B x A) ↔ ¬ x(x = B x A))
9 df-clel 2033 . . 3 (B Ax(x = B x A))
108, 9xchbinxr 607 . 2 (x ¬ (x = B x A) ↔ ¬ B A)
111, 7, 103bitri 195 1 ((A ∩ {B}) = ∅ ↔ ¬ B A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  cin 2910  c0 3218  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219  df-sn 3373
This theorem is referenced by:  disjsn2  3424  ndmima  4645  funtpg  4893  fnunsn  4949  ressnop0  5287  ftpg  5290  fsnunf  5305  fsnunfv  5306  fzpreddisj  8683  fzp1disj  8692  frecfzennn  8864
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