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| Mirrors > Home > MPE Home > Th. List > cfon | Structured version Visualization version GIF version | ||
| Description: The cofinality of any set is an ordinal (although it only makes sense when 𝐴 is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.) |
| Ref | Expression |
|---|---|
| cfon | ⊢ (cf‘𝐴) ∈ On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cardcf 10292 | . 2 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴) | |
| 2 | cardon 9984 | . 2 ⊢ (card‘(cf‘𝐴)) ∈ On | |
| 3 | 1, 2 | eqeltrri 2838 | 1 ⊢ (cf‘𝐴) ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Oncon0 6384 ‘cfv 6561 cardccrd 9975 cfccf 9977 |
| 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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-card 9979 df-cf 9981 |
| This theorem is referenced by: cfslb2n 10308 cfsmolem 10310 cfcoflem 10312 cfcof 10314 cfidm 10315 alephreg 10622 winaon 10728 inawina 10730 winainf 10734 rankcf 10817 tskcard 10821 gruina 10858 |
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