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Mirrors > Home > MPE Home > Th. List > oncardid | Structured version Visualization version GIF version |
Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10565, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.) |
Ref | Expression |
---|---|
oncardid | ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onenon 9967 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | |
2 | cardid2 9971 | . 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5143 dom cdm 5673 Oncon0 6364 ‘cfv 6543 ≈ cen 8955 cardccrd 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-en 8959 df-card 9957 |
This theorem is referenced by: cardom 10004 alephinit 10113 dfac12k 10165 harval3 42959 |
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