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}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)

Proof of Theorem disjsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj1 3330	. 2 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}))
2		con2b 628	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴))
3		velsn 3458	. . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	3	imbi1i 236	. . . 4 ⊢ ((𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴))
5		imnan 659	. . . 4 ⊢ ((𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴))
6	2, 4, 5	3bitri 204	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴))
7	6	albii 1404	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴))
8		alnex 1433	. . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴))
9		df-clel 2084	. . 3 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴))
10	8, 9	xchbinxr 643	. 2 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ 𝐵 ∈ 𝐴)
11	1, 7, 10	3bitri 204	1 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)

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