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| Mirrors > Home > MPE Home > Th. List > cardon | Structured version Visualization version GIF version | ||
| Description: The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.) |
| Ref | Expression |
|---|---|
| cardon | ⊢ (card‘𝐴) ∈ On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cardf2 9983 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
| 2 | 0elon 6438 | . 2 ⊢ ∅ ∈ On | |
| 3 | 1, 2 | f0cli 7118 | 1 ⊢ (card‘𝐴) ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 {cab 2714 ∃wrex 3070 class class class wbr 5143 Oncon0 6384 ‘cfv 6561 ≈ 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-pr 5432 |
| 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-card 9979 |
| This theorem is referenced by: isnum3 9994 cardidm 9999 ficardom 10001 cardne 10005 carden2b 10007 cardlim 10012 cardsdomelir 10013 cardsdomel 10014 iscard 10015 iscard2 10016 carddom2 10017 carduni 10021 cardom 10026 cardsdom2 10028 domtri2 10029 cardval2 10031 infxpidm2 10057 dfac8b 10071 numdom 10078 indcardi 10081 alephnbtwn 10111 alephnbtwn2 10112 alephsucdom 10119 cardaleph 10129 iscard3 10133 alephinit 10135 alephsson 10140 alephval3 10150 dfac12r 10187 dfac12k 10188 cardadju 10235 djunum 10236 pwsdompw 10243 cff 10288 cardcf 10292 cfon 10295 cfeq0 10296 cfsuc 10297 cff1 10298 cfflb 10299 cflim2 10303 cfss 10305 fin1a2lem9 10448 ttukeylem6 10554 ttukeylem7 10555 unsnen 10593 inar1 10815 tskcard 10821 tskuni 10823 gruina 10858 iscard4 43546 minregex 43547 |
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