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 9361 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
2 | 0elon 6238 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6857 | 1 ⊢ (card‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 {cab 2799 ∃wrex 3139 class class class wbr 5058 Oncon0 6185 ‘cfv 6349 ≈ cen 8495 cardccrd 9353 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-card 9357 |
This theorem is referenced by: isnum3 9372 cardidm 9377 ficardom 9379 cardne 9383 carden2b 9385 cardlim 9390 cardsdomelir 9391 cardsdomel 9392 iscard 9393 iscard2 9394 carddom2 9395 carduni 9399 cardom 9404 cardsdom2 9406 domtri2 9407 cardval2 9409 infxpidm2 9432 dfac8b 9446 numdom 9453 indcardi 9456 alephnbtwn 9486 alephnbtwn2 9487 alephsucdom 9494 cardaleph 9504 iscard3 9508 alephinit 9510 alephsson 9515 alephval3 9525 dfac12r 9561 dfac12k 9562 cardadju 9609 djunum 9610 pwsdompw 9615 cff 9659 cardcf 9663 cfon 9666 cfeq0 9667 cfsuc 9668 cff1 9669 cfflb 9670 cflim2 9674 cfss 9676 fin1a2lem9 9819 ttukeylem6 9925 ttukeylem7 9926 unsnen 9964 inar1 10186 tskcard 10192 tskuni 10194 gruina 10229 iscard4 39780 |
Copyright terms: Public domain | W3C validator |