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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 8769 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
2 | 0elon 5778 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6370 | 1 ⊢ (card‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 {cab 2608 ∃wrex 2913 class class class wbr 4653 Oncon0 5723 ‘cfv 5888 ≈ cen 7952 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-card 8765 |
This theorem is referenced by: isnum3 8780 cardidm 8785 ficardom 8787 cardne 8791 carden2b 8793 cardlim 8798 cardsdomelir 8799 cardsdomel 8800 iscard 8801 iscard2 8802 carddom2 8803 carduni 8807 cardom 8812 cardsdom2 8814 domtri2 8815 cardval2 8817 infxpidm2 8840 dfac8b 8854 numdom 8861 indcardi 8864 alephnbtwn 8894 alephnbtwn2 8895 alephsucdom 8902 cardaleph 8912 iscard3 8916 alephinit 8918 alephsson 8923 alephval3 8933 dfac12r 8968 dfac12k 8969 cardacda 9020 cdanum 9021 pwsdompw 9026 cff 9070 cardcf 9074 cfon 9077 cfeq0 9078 cfsuc 9079 cff1 9080 cfflb 9081 cflim2 9085 cfss 9087 fin1a2lem9 9230 ttukeylem6 9336 ttukeylem7 9337 unsnen 9375 inar1 9597 tskcard 9603 tskuni 9605 gruina 9640 mreexexdOLD 16309 |
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