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Theorem disjsn 3786
 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 3593 . 2 ((A ∩ {B}) = x(x A → ¬ x {B}))
2 con2b 324 . . . 4 ((x A → ¬ x {B}) ↔ (x {B} → ¬ x A))
3 elsn 3748 . . . . 5 (x {B} ↔ x = B)
43imbi1i 315 . . . 4 ((x {B} → ¬ x A) ↔ (x = B → ¬ x A))
5 imnan 411 . . . 4 ((x = B → ¬ x A) ↔ ¬ (x = B x A))
62, 4, 53bitri 262 . . 3 ((x A → ¬ x {B}) ↔ ¬ (x = B x A))
76albii 1566 . 2 (x(x A → ¬ x {B}) ↔ x ¬ (x = B x A))
8 alnex 1543 . . 3 (x ¬ (x = B x A) ↔ ¬ x(x = B x A))
9 df-clel 2349 . . 3 (B Ax(x = B x A))
108, 9xchbinxr 302 . 2 (x ¬ (x = B x A) ↔ ¬ B A)
111, 7, 103bitri 262 1 ((A ∩ {B}) = ↔ ¬ B A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∅c0 3550  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-sn 3741 This theorem is referenced by:  disjsn2  3787  ssunsn2  3865  dfiota4  4372  elsuc  4413  nndisjeq  4429  vfinncsp  4554  phiall  4618  ndmima  5025  fnunsn  5190  ressnop0  5436  1p1e2c  6155  2p1e3c  6156  nchoicelem7  6295  nchoicelem14  6302
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