![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9663 | . 2 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴) | |
2 | cardon 9357 | . 2 ⊢ (card‘(cf‘𝐴)) ∈ On | |
3 | 1, 2 | eqeltrri 2887 | 1 ⊢ (cf‘𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Oncon0 6159 ‘cfv 6324 cardccrd 9348 cfccf 9350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-card 9352 df-cf 9354 |
This theorem is referenced by: cfslb2n 9679 cfsmolem 9681 cfcoflem 9683 cfcof 9685 cfidm 9686 alephreg 9993 winaon 10099 inawina 10101 winainf 10105 rankcf 10188 tskcard 10192 gruina 10229 |
Copyright terms: Public domain | W3C validator |