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| Mirrors > Home > MPE Home > Th. List > orddisj | Structured version Visualization version 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 6402 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
| 2 | disjsn 4711 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
| 3 | 1, 2 | sylibr 234 | 1 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 {csn 4626 Ord word 6383 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 df-we 5639 df-ord 6387 |
| This theorem is referenced by: orddif 6480 omsucne 7906 tfrlem10 8427 enrefnn 9087 pssnn 9208 unfi 9211 phplem2OLD 9255 isinf 9296 isinfOLD 9297 dif1ennnALT 9311 ackbij1lem5 10263 ackbij1lem14 10272 ackbij1lem16 10274 unsnen 10593 pwfi2f1o 43108 |
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