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Mirrors > Home > ILE Home > Th. List > disjsn | GIF version |
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.) |
Ref | Expression |
---|---|
disjsn | ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj1 3294 | . 2 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵})) | |
2 | con2b 625 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴)) | |
3 | velsn 3415 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 3 | imbi1i 236 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
5 | imnan 656 | . . . 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 1399 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) |
8 | alnex 1428 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
9 | df-clel 2077 | . . 3 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
10 | 8, 9 | xchbinxr 640 | . 2 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ 𝐵 ∈ 𝐴) |
11 | 1, 7, 10 | 3bitri 204 | 1 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∩ cin 2972 ∅c0 3251 {csn 3398 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-dif 2975 df-in 2979 df-nul 3252 df-sn 3404 |
This theorem is referenced by: disjsn2 3455 orddisj 4289 ndmima 4722 funtpg 4970 fnunsn 5026 ressnop0 5365 ftpg 5368 fsnunf 5383 fsnunfv 5384 phpm 6351 fiunsnnn 6365 ac6sfi 6379 unsnfi 6384 pm54.43 6459 fzpreddisj 9088 fzp1disj 9097 frecfzennn 9419 |
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