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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 8652 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
2 | 0elon 5695 | . 2 ⊢ ∅ ∈ On | |
3 | 1, 2 | f0cli 6278 | 1 ⊢ (card‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {cab 2596 ∃wrex 2897 class class class wbr 4583 Oncon0 5640 ‘cfv 5804 ≈ cen 7838 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-card 8648 |
This theorem is referenced by: isnum3 8663 cardidm 8668 ficardom 8670 cardne 8674 carden2b 8676 cardlim 8681 cardsdomelir 8682 cardsdomel 8683 iscard 8684 iscard2 8685 carddom2 8686 carduni 8690 cardom 8695 cardsdom2 8697 domtri2 8698 cardval2 8700 infxpidm2 8723 dfac8b 8737 numdom 8744 indcardi 8747 alephnbtwn 8777 alephnbtwn2 8778 alephsucdom 8785 cardaleph 8795 iscard3 8799 alephinit 8801 alephsson 8806 alephval3 8816 dfac12r 8851 dfac12k 8852 cardacda 8903 cdanum 8904 pwsdompw 8909 cff 8953 cardcf 8957 cfon 8960 cfeq0 8961 cfsuc 8962 cff1 8963 cfflb 8964 cflim2 8968 cfss 8970 fin1a2lem9 9113 ttukeylem6 9219 ttukeylem7 9220 unsnen 9254 inar1 9476 tskcard 9482 tskuni 9484 gruina 9519 mreexexdOLD 16132 |
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