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Mirrors > Home > ILE Home > Th. List > nelsn | 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 3599 | . 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
2 | 1 | necon3ai 2389 | 1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2141 ≠ wne 2340 {csn 3581 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-sn 3587 |
This theorem is referenced by: nnoddn2prmb 12203 |
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