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Theorem cfon 7926
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 )  e.  On

Proof of Theorem cfon
StepHypRef Expression
1 cardcf 7923 . 2  |-  ( card `  ( cf `  A
) )  =  ( cf `  A )
2 cardon 7622 . 2  |-  ( card `  ( cf `  A
) )  e.  On
31, 2eqeltrri 2387 1  |-  ( cf `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1701   Oncon0 4429   ` cfv 5292   cardccrd 7613   cfccf 7615
This theorem is referenced by:  cfslb2n  7939  cfsmolem  7941  cfcoflem  7943  cfcof  7945  cfidm  7946  alephreg  8249  winaon  8355  inawina  8357  winainf  8361  rankcf  8444  tskcard  8448  gruina  8485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-er 6702  df-en 6907  df-card 7617  df-cf 7619
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