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Theorem cardon 7577
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 7576 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4445 . 2  |-  (/)  e.  On
31, 2f0cli 5671 1  |-  ( card `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   {cab 2269   E.wrex 2544   class class class wbr 4023   Oncon0 4392   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  isnum3  7587  cardidm  7592  ficardom  7594  cardne  7598  carden2b  7600  cardlim  7605  cardsdomelir  7606  cardsdomel  7607  iscard  7608  iscard2  7609  carddom2  7610  carduni  7614  cardom  7619  cardsdom2  7621  domtri2  7622  cardval2  7624  infxpidm2  7644  dfac8b  7658  numdom  7665  indcardi  7668  alephnbtwn  7698  alephnbtwn2  7699  alephsucdom  7706  cardaleph  7716  iscard3  7720  alephinit  7722  alephsson  7727  alephval3  7737  dfac12r  7772  dfac12k  7773  cardacda  7824  cdanum  7825  pwsdompw  7830  cff  7874  cardcf  7878  cfon  7881  cfeq0  7882  cfsuc  7883  cff1  7884  cfflb  7885  cflim2  7889  cfss  7891  fin1a2lem9  8034  ttukeylem6  8141  ttukeylem7  8142  unsnen  8175  inar1  8397  tskcard  8403  tskuni  8405  gruina  8440  mreexexd  13550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-card 7572
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