Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > disjsn2 | Structured version Visualization version 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 4142 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2616 | . . 3 ⊢ (𝐵 ∈ {𝐴} → 𝐴 = 𝐵) |
3 | 2 | necon3ai 2807 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
4 | disjsn 4192 | . 2 ⊢ (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴}) | |
5 | 3, 4 | sylibr 223 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 df-sn 4126 |
This theorem is referenced by: disjpr2 4194 disjpr2OLD 4195 difprsn1 4271 diftpsn3OLD 4274 otsndisj 4904 xpsndisj 5476 funprg 5854 funprgOLD 5855 funtp 5859 funcnvpr 5864 f1oprg 6093 phplem1 8024 pm54.43 8709 pr2nelem 8710 f1oun2prg 13512 s3sndisj 13554 sumpr 14321 cshwsdisj 15643 setsfun0 15726 setscom 15731 xpsc0 16043 xpsc1 16044 dmdprdpr 18271 dprdpr 18272 ablfac1eulem 18294 m2detleib 20256 dishaus 20996 dissnlocfin 21142 xpstopnlem1 21422 perfectlem2 24755 esumpr 29455 esum2dlem 29481 onint1 31618 bj-disjsn01 32130 poimirlem26 32605 sumpair 38217 perfectALTVlem2 40165 gsumpr 41932 |
Copyright terms: Public domain | W3C validator |