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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 3418 | . 2 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵})) | |
2 | con2b 659 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴)) | |
3 | velsn 3549 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 3 | imbi1i 237 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴)) |
5 | imnan 680 | . . . 4 ⊢ ((𝑥 = 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
6 | 2, 4, 5 | 3bitri 205 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) |
7 | 6 | albii 1447 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵}) ↔ ∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) |
8 | alnex 1476 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
9 | df-clel 2136 | . . 3 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
10 | 8, 9 | xchbinxr 673 | . 2 ⊢ (∀𝑥 ¬ (𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ ¬ 𝐵 ∈ 𝐴) |
11 | 1, 7, 10 | 3bitri 205 | 1 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∩ cin 3075 ∅c0 3368 {csn 3532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-dif 3078 df-in 3082 df-nul 3369 df-sn 3538 |
This theorem is referenced by: disjsn2 3594 ssdifsn 3659 orddisj 4469 ndmima 4924 funtpg 5182 fnunsn 5238 ressnop0 5609 ftpg 5612 fsnunf 5628 fsnunfv 5629 enpr2d 6719 phpm 6767 fiunsnnn 6783 ac6sfi 6800 unsnfi 6815 tpfidisj 6824 iunfidisj 6842 pm54.43 7063 dju1en 7086 fzpreddisj 9882 fzp1disj 9891 frecfzennn 10230 hashunsng 10585 hashxp 10604 fsumsplitsn 11211 sumtp 11215 fsumsplitsnun 11220 fsum2dlemstep 11235 fsumconst 11255 fsumabs 11266 fsumiun 11278 ennnfonelemhf1o 11962 structcnvcnv 12014 fsumcncntop 12764 |
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