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Mirrors > Home > ILE Home > Th. List > nordeq | GIF version |
Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
nordeq | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 4285 | . . . 4 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | eleq1 2141 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
3 | 2 | notbid 624 | . . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴)) |
4 | 1, 3 | syl5ibcom 153 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴)) |
5 | 4 | necon2ad 2302 | . 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐴 ≠ 𝐵)) |
6 | 5 | imp 122 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 Ord word 4117 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-sn 3404 |
This theorem is referenced by: phplem1 6338 |
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