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Theorem cardon 7573
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem cardon
StepHypRef Expression
1 cardf2 7572 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4445 . 2  |-  (/)  e.  On
31, 2f0cli 5633 1  |-  ( card `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1685   {cab 2271   E.wrex 2546   class class class wbr 4025   Oncon0 4392   ` cfv 5222    ~~ cen 6856   cardccrd 7564
This theorem is referenced by:  isnum3  7583  cardidm  7588  ficardom  7590  cardne  7594  carden2b  7596  cardlim  7601  cardsdomelir  7602  cardsdomel  7603  iscard  7604  iscard2  7605  carddom2  7606  carduni  7610  cardom  7615  cardsdom2  7617  domtri2  7618  cardval2  7620  infxpidm2  7640  dfac8b  7654  numdom  7661  indcardi  7664  alephnbtwn  7694  alephnbtwn2  7695  alephsucdom  7702  cardaleph  7712  iscard3  7716  alephinit  7718  alephsson  7723  alephval3  7733  dfac12r  7768  dfac12k  7769  cardacda  7820  cdanum  7821  pwsdompw  7826  cff  7870  cardcf  7874  cfon  7877  cfeq0  7878  cfsuc  7879  cff1  7880  cfflb  7881  cflim2  7885  cfss  7887  fin1a2lem9  8030  ttukeylem6  8137  ttukeylem7  8138  unsnen  8171  inar1  8393  tskcard  8399  tskuni  8401  gruina  8436  mreexexd  13545
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-fv 5230  df-card 7568
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