Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nelsn | Structured version Visualization version GIF version |
Description: If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.) |
Ref | Expression |
---|---|
nelsn | ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsni 4576 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
2 | 1 | necon3ai 3041 | 1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 ≠ wne 3016 {csn 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-sn 4560 |
This theorem is referenced by: fvunsn 6934 nnoddn2prmb 16140 lbsextlem4 19864 cnfldfunALT 20488 obslbs 20804 logbgcd1irr 25299 upgrres1 27023 cycpmco2 30703 lindssn 30867 submateqlem1 30972 submateqlem2 30973 qqhval2 31123 derangsn 32315 prjspersym 39137 prjspreln0 39139 prjspvs 39140 pr2eldif1 39793 pr2eldif2 39794 clsk3nimkb 40270 clsk1indlem1 40275 disjf1o 41332 cnrefiisplem 41990 fperdvper 42083 dvnmul 42108 wallispi 42236 etransc 42449 gsumge0cl 42534 meadjiunlem 42628 hspmbllem2 42790 |
Copyright terms: Public domain | W3C validator |