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Theorem cardon 7764
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 7763 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4575 . 2  |-  (/)  e.  On
31, 2f0cli 5819 1  |-  ( card `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   {cab 2373   E.wrex 2650   class class class wbr 4153   Oncon0 4522   ` cfv 5394    ~~ cen 7042   cardccrd 7755
This theorem is referenced by:  isnum3  7774  cardidm  7779  ficardom  7781  cardne  7785  carden2b  7787  cardlim  7792  cardsdomelir  7793  cardsdomel  7794  iscard  7795  iscard2  7796  carddom2  7797  carduni  7801  cardom  7806  cardsdom2  7808  domtri2  7809  cardval2  7811  infxpidm2  7831  dfac8b  7845  numdom  7852  indcardi  7855  alephnbtwn  7885  alephnbtwn2  7886  alephsucdom  7893  cardaleph  7903  iscard3  7907  alephinit  7909  alephsson  7914  alephval3  7924  dfac12r  7959  dfac12k  7960  cardacda  8011  cdanum  8012  pwsdompw  8017  cff  8061  cardcf  8065  cfon  8068  cfeq0  8069  cfsuc  8070  cff1  8071  cfflb  8072  cflim2  8076  cfss  8078  fin1a2lem9  8221  ttukeylem6  8327  ttukeylem7  8328  unsnen  8361  inar1  8583  tskcard  8589  tskuni  8591  gruina  8626  mreexexd  13800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-card 7759
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