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Theorem cfon 9365
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.)
Assertion
Ref Expression
cfon (cf‘𝐴) ∈ On

Proof of Theorem cfon
StepHypRef Expression
1 cardcf 9362 . 2 (card‘(cf‘𝐴)) = (cf‘𝐴)
2 cardon 9056 . 2 (card‘(cf‘𝐴)) ∈ On
31, 2eqeltrri 2875 1 (cf‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  Oncon0 5941  cfv 6101  cardccrd 9047  cfccf 9049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-er 7982  df-en 8196  df-card 9051  df-cf 9053
This theorem is referenced by:  cfslb2n  9378  cfsmolem  9380  cfcoflem  9382  cfcof  9384  cfidm  9385  alephreg  9692  winaon  9798  inawina  9800  winainf  9804  rankcf  9887  tskcard  9891  gruina  9928
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