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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 6076 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | disjsn 4548 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | sylibr 235 | 1 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1520 ∈ wcel 2079 ∩ cin 3853 ∅c0 4206 {csn 4466 Ord word 6057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-opab 5019 df-eprel 5345 df-fr 5394 df-we 5396 df-ord 6061 |
This theorem is referenced by: orddif 6151 omsucne 7445 tfrlem10 7866 phplem2 8534 isinf 8567 pssnn 8572 dif1en 8587 ackbij1lem5 9481 ackbij1lem14 9490 ackbij1lem16 9492 unsnen 9810 pwfi2f1o 39132 |
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