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Mirrors > Home > ILE Home > Th. List > orddisj | GIF version |
Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
orddisj | ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 4457 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | disjsn 3585 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | sylibr 133 | 1 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 ∩ cin 3070 ∅c0 3363 {csn 3527 Ord word 4284 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-nul 3364 df-sn 3533 |
This theorem is referenced by: orddif 4462 phplem2 6747 ennnfonelemhf1o 11926 |
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