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Mirrors > Home > ILE Home > Th. List > disjsn2 | GIF version |
Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
disjsn2 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 3545 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2145 | . . 3 ⊢ (𝐵 ∈ {𝐴} → 𝐴 = 𝐵) |
3 | 2 | necon3ai 2357 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
4 | disjsn 3585 | . 2 ⊢ (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴}) | |
5 | 3, 4 | sylibr 133 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∩ cin 3070 ∅c0 3363 {csn 3527 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-nul 3364 df-sn 3533 |
This theorem is referenced by: disjpr2 3587 difprsn1 3659 diftpsn3 3661 xpsndisj 4965 funprg 5173 funtp 5176 f1oprg 5411 xp01disjl 6331 enpr2d 6711 phplem1 6746 prfidisj 6815 djuinr 6948 pm54.43 7046 pr2nelem 7047 sumpr 11182 setsfun0 11995 setscom 11999 |
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