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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 3315 | . 2 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵})) | |
2 | con2b 626 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴)) | |
3 | velsn 3439 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 3 | imbi1i 236 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
5 | imnan 657 | . . . 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 1400 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) |
8 | alnex 1429 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
9 | df-clel 2079 | . . 3 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
10 | 8, 9 | xchbinxr 641 | . 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 1283 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∩ cin 2983 ∅c0 3269 {csn 3422 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-v 2614 df-dif 2986 df-in 2990 df-nul 3270 df-sn 3428 |
This theorem is referenced by: disjsn2 3479 ssdifsn 3542 orddisj 4325 ndmima 4764 funtpg 5018 fnunsn 5074 ressnop0 5420 ftpg 5423 fsnunf 5438 fsnunfv 5439 phpm 6511 fiunsnnn 6527 ac6sfi 6544 unsnfi 6556 pm54.43 6721 fzpreddisj 9378 fzp1disj 9387 frecfzennn 9722 hashunsng 10050 hashxp 10069 |
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