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Theorem cardon 9361
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 9360 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 0elon 6237 . 2 ∅ ∈ On
31, 2f0cli 6856 1 (card‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  {cab 2796  wrex 3136   class class class wbr 5057  Oncon0 6184  cfv 6348  cen 8494  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-card 9356
This theorem is referenced by:  isnum3  9371  cardidm  9376  ficardom  9378  cardne  9382  carden2b  9384  cardlim  9389  cardsdomelir  9390  cardsdomel  9391  iscard  9392  iscard2  9393  carddom2  9394  carduni  9398  cardom  9403  cardsdom2  9405  domtri2  9406  cardval2  9408  infxpidm2  9431  dfac8b  9445  numdom  9452  indcardi  9455  alephnbtwn  9485  alephnbtwn2  9486  alephsucdom  9493  cardaleph  9503  iscard3  9507  alephinit  9509  alephsson  9514  alephval3  9524  dfac12r  9560  dfac12k  9561  cardadju  9608  djunum  9609  pwsdompw  9614  cff  9658  cardcf  9662  cfon  9665  cfeq0  9666  cfsuc  9667  cff1  9668  cfflb  9669  cflim2  9673  cfss  9675  fin1a2lem9  9818  ttukeylem6  9924  ttukeylem7  9925  unsnen  9963  inar1  10185  tskcard  10191  tskuni  10193  gruina  10228  iscard4  39778
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