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Theorem cardon 8722
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 8721 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 0elon 5742 . 2 ∅ ∈ On
31, 2f0cli 6331 1 (card‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  {cab 2607  wrex 2908   class class class wbr 4618  Oncon0 5687  cfv 5852  cen 7904  cardccrd 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-card 8717
This theorem is referenced by:  isnum3  8732  cardidm  8737  ficardom  8739  cardne  8743  carden2b  8745  cardlim  8750  cardsdomelir  8751  cardsdomel  8752  iscard  8753  iscard2  8754  carddom2  8755  carduni  8759  cardom  8764  cardsdom2  8766  domtri2  8767  cardval2  8769  infxpidm2  8792  dfac8b  8806  numdom  8813  indcardi  8816  alephnbtwn  8846  alephnbtwn2  8847  alephsucdom  8854  cardaleph  8864  iscard3  8868  alephinit  8870  alephsson  8875  alephval3  8885  dfac12r  8920  dfac12k  8921  cardacda  8972  cdanum  8973  pwsdompw  8978  cff  9022  cardcf  9026  cfon  9029  cfeq0  9030  cfsuc  9031  cff1  9032  cfflb  9033  cflim2  9037  cfss  9039  fin1a2lem9  9182  ttukeylem6  9288  ttukeylem7  9289  unsnen  9327  inar1  9549  tskcard  9555  tskuni  9557  gruina  9592  mreexexdOLD  16241
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