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Theorem cardon 9370
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 9369 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 0elon 6231 . 2 ∅ ∈ On
31, 2f0cli 6855 1 (card‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  {cab 2802  wrex 3134   class class class wbr 5052  Oncon0 6178  cfv 6343  cen 8502  cardccrd 9361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-card 9365
This theorem is referenced by:  isnum3  9380  cardidm  9385  ficardom  9387  cardne  9391  carden2b  9393  cardlim  9398  cardsdomelir  9399  cardsdomel  9400  iscard  9401  iscard2  9402  carddom2  9403  carduni  9407  cardom  9412  cardsdom2  9414  domtri2  9415  cardval2  9417  infxpidm2  9441  dfac8b  9455  numdom  9462  indcardi  9465  alephnbtwn  9495  alephnbtwn2  9496  alephsucdom  9503  cardaleph  9513  iscard3  9517  alephinit  9519  alephsson  9524  alephval3  9534  dfac12r  9570  dfac12k  9571  cardadju  9618  djunum  9619  pwsdompw  9624  cff  9668  cardcf  9672  cfon  9675  cfeq0  9676  cfsuc  9677  cff1  9678  cfflb  9679  cflim2  9683  cfss  9685  fin1a2lem9  9828  ttukeylem6  9934  ttukeylem7  9935  unsnen  9973  inar1  10195  tskcard  10201  tskuni  10203  gruina  10238  iscard4  40157
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