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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 6202 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | disjsn 4639 | . 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 1528 ∈ wcel 2105 ∩ cin 3932 ∅c0 4288 {csn 4557 Ord word 6183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 df-fr 5507 df-we 5509 df-ord 6187 |
This theorem is referenced by: orddif 6277 omsucne 7587 tfrlem10 8012 phplem2 8685 isinf 8719 pssnn 8724 dif1en 8739 ackbij1lem5 9634 ackbij1lem14 9643 ackbij1lem16 9645 unsnen 9963 pwfi2f1o 39574 |
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