![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > disjsn2 | Structured version Visualization version GIF version |
Description: Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
disjsn2 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4647 | . . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2740 | . . 3 ⊢ (𝐵 ∈ {𝐴} → 𝐴 = 𝐵) |
3 | 2 | necon3ai 2962 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
4 | disjsn 4715 | . 2 ⊢ (({𝐴} ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ {𝐴}) | |
5 | 3, 4 | sylibr 234 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∩ cin 3961 ∅c0 4338 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-v 3479 df-dif 3965 df-in 3969 df-nul 4339 df-sn 4631 |
This theorem is referenced by: disjpr2 4717 disjtpsn 4719 difprsn1 4804 otsndisj 5528 xpsndisj 6184 funprg 6621 funtp 6624 funcnvpr 6629 f1oprg 6893 xp01disjl 8528 enpr2dOLD 9088 phplem1OLD 9251 djuin 9955 pm54.43 10038 pr2nelemOLD 10040 f1oun2prg 14952 s3sndisj 15002 sumpr 15780 cshwsdisj 17132 setsfun0 17205 setscom 17213 gsumpr 19987 dmdprdpr 20083 dprdpr 20084 ablfac1eulem 20106 cnfldfunALT 21396 cnfldfunALTOLD 21409 cnfldfunALTOLDOLD 21410 m2detleib 22652 dishaus 23405 dissnlocfin 23552 xpstopnlem1 23832 perfectlem2 27288 cosnopne 32708 prodpr 32832 esumpr 34046 esum2dlem 34072 prodfzo03 34596 onint1 36431 bj-disjsn01 36934 lindsadd 37599 poimirlem26 37632 sumpair 44972 perfectALTVlem2 47646 |
Copyright terms: Public domain | W3C validator |