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Theorem cardon 8316
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 )  e.  On

Proof of Theorem cardon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 8315 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4920 . 2  |-  (/)  e.  On
31, 2f0cli 6018 1  |-  ( card `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   {cab 2439   E.wrex 2805   class class class wbr 4439   Oncon0 4867   ` cfv 5570    ~~ cen 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-card 8311
This theorem is referenced by:  isnum3  8326  cardidm  8331  ficardom  8333  cardne  8337  carden2b  8339  cardlim  8344  cardsdomelir  8345  cardsdomel  8346  iscard  8347  iscard2  8348  carddom2  8349  carduni  8353  cardom  8358  cardsdom2  8360  domtri2  8361  cardval2  8363  infxpidm2  8385  dfac8b  8403  numdom  8410  indcardi  8413  alephnbtwn  8443  alephnbtwn2  8444  alephsucdom  8451  cardaleph  8461  iscard3  8465  alephinit  8467  alephsson  8472  alephval3  8482  dfac12r  8517  dfac12k  8518  cardacda  8569  cdanum  8570  pwsdompw  8575  cff  8619  cardcf  8623  cfon  8626  cfeq0  8627  cfsuc  8628  cff1  8629  cfflb  8630  cflim2  8634  cfss  8636  fin1a2lem9  8779  ttukeylem6  8885  ttukeylem7  8886  unsnen  8919  inar1  9142  tskcard  9148  tskuni  9150  gruina  9185  mreexexd  15137
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