![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9935 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
2 | 0elon 6416 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 7097 | 1 ⊢ (card‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 {cab 2710 ∃wrex 3071 class class class wbr 5148 Oncon0 6362 ‘cfv 6541 ≈ cen 8933 cardccrd 9927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6365 df-on 6366 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-card 9931 |
This theorem is referenced by: isnum3 9946 cardidm 9951 ficardom 9953 cardne 9957 carden2b 9959 cardlim 9964 cardsdomelir 9965 cardsdomel 9966 iscard 9967 iscard2 9968 carddom2 9969 carduni 9973 cardom 9978 cardsdom2 9980 domtri2 9981 cardval2 9983 infxpidm2 10009 dfac8b 10023 numdom 10030 indcardi 10033 alephnbtwn 10063 alephnbtwn2 10064 alephsucdom 10071 cardaleph 10081 iscard3 10085 alephinit 10087 alephsson 10092 alephval3 10102 dfac12r 10138 dfac12k 10139 cardadju 10186 djunum 10187 pwsdompw 10196 cff 10240 cardcf 10244 cfon 10247 cfeq0 10248 cfsuc 10249 cff1 10250 cfflb 10251 cflim2 10255 cfss 10257 fin1a2lem9 10400 ttukeylem6 10506 ttukeylem7 10507 unsnen 10545 inar1 10767 tskcard 10773 tskuni 10775 gruina 10810 iscard4 42270 minregex 42271 |
Copyright terms: Public domain | W3C validator |