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Theorem cardon 8653
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.)
Assertion
Ref Expression
cardon (card‘𝐴) ∈ On

Proof of Theorem cardon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 8652 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 0elon 5695 . 2 ∅ ∈ On
31, 2f0cli 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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