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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 8529 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴) | |
2 | isnumi 9363 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → 𝐴 ∈ dom card) | |
3 | 1, 2 | mpdan 683 | 1 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 Oncon0 6184 ≈ cen 8494 cardccrd 9352 |
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-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-en 8498 df-card 9356 |
This theorem is referenced by: oncardval 9372 oncardid 9373 cardnn 9380 iscard 9392 carduni 9398 nnsdomel 9407 harsdom 9412 pm54.43lem 9416 infxpenlem 9427 infxpidm2 9431 onssnum 9454 alephnbtwn 9485 alephnbtwn2 9486 alephordilem1 9487 alephord2 9490 alephsdom 9500 cardaleph 9503 infenaleph 9505 alephinit 9509 iunfictbso 9528 ficardun2 9613 pwsdompw 9614 infunsdom1 9623 ackbij2 9653 cfflb 9669 sdom2en01 9712 fin23lem22 9737 iunctb 9984 alephadd 9987 alephmul 9988 alephexp1 9989 alephsuc3 9990 canthp1lem2 10063 pwfseqlem4a 10071 pwfseqlem4 10072 pwfseqlem5 10073 gchaleph 10081 gchaleph2 10082 hargch 10083 cygctb 18941 ttac 39511 numinfctb 39581 isnumbasgrplem2 39582 isnumbasabl 39584 iscard4 39778 harsucnn 39781 harval3 39782 harval3on 39783 aleph1min 39794 |
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