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| Mirrors > Home > MPE Home > Th. List > onenon | Structured version Visualization version GIF version | ||
| Description: Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| Ref | Expression |
|---|---|
| onenon | ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enrefg 9024 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
| 2 | isnumi 9986 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → 𝐴 ∈ dom card) | |
| 3 | 1, 2 | mpdan 687 | 1 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 Oncon0 6384 ≈ cen 8982 cardccrd 9975 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 df-card 9979 |
| This theorem is referenced by: oncardval 9995 oncardid 9996 cardnn 10003 iscard 10015 carduni 10021 nnsdomel 10030 harsdom 10035 harsucnn 10038 pm54.43lem 10040 infxpenlem 10053 infxpidm2 10057 onssnum 10080 alephnbtwn 10111 alephnbtwn2 10112 alephordilem1 10113 alephord2 10116 alephsdom 10126 cardaleph 10129 infenaleph 10131 alephinit 10135 iunfictbso 10154 ficardun2 10242 pwsdompw 10243 infunsdom1 10252 ackbij2 10282 cfflb 10299 sdom2en01 10342 fin23lem22 10367 iunctb 10614 alephadd 10617 alephmul 10618 alephexp1 10619 alephsuc3 10620 canthp1lem2 10693 pwfseqlem4a 10701 pwfseqlem4 10702 pwfseqlem5 10703 gchaleph 10711 gchaleph2 10712 hargch 10713 cygctb 19910 ttac 43048 numinfctb 43115 isnumbasgrplem2 43116 isnumbasabl 43118 iscard4 43546 minregex2 43548 harval3 43551 harval3on 43552 aleph1min 43570 |
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