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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 6209 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | disjsn 4647 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | sylibr 236 | 1 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ∅c0 4291 {csn 4567 Ord word 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-fr 5514 df-we 5516 df-ord 6194 |
This theorem is referenced by: orddif 6284 omsucne 7598 tfrlem10 8023 phplem2 8697 isinf 8731 pssnn 8736 dif1en 8751 ackbij1lem5 9646 ackbij1lem14 9655 ackbij1lem16 9657 unsnen 9975 pwfi2f1o 39716 |
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