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Theorem cardon 7593
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 7592 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4461 . 2  |-  (/)  e.  On
31, 2f0cli 5687 1  |-  ( card `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   {cab 2282   E.wrex 2557   class class class wbr 4039   Oncon0 4408   ` cfv 5271    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  isnum3  7603  cardidm  7608  ficardom  7610  cardne  7614  carden2b  7616  cardlim  7621  cardsdomelir  7622  cardsdomel  7623  iscard  7624  iscard2  7625  carddom2  7626  carduni  7630  cardom  7635  cardsdom2  7637  domtri2  7638  cardval2  7640  infxpidm2  7660  dfac8b  7674  numdom  7681  indcardi  7684  alephnbtwn  7714  alephnbtwn2  7715  alephsucdom  7722  cardaleph  7732  iscard3  7736  alephinit  7738  alephsson  7743  alephval3  7753  dfac12r  7788  dfac12k  7789  cardacda  7840  cdanum  7841  pwsdompw  7846  cff  7890  cardcf  7894  cfon  7897  cfeq0  7898  cfsuc  7899  cff1  7900  cfflb  7901  cflim2  7905  cfss  7907  fin1a2lem9  8050  ttukeylem6  8157  ttukeylem7  8158  unsnen  8191  inar1  8413  tskcard  8419  tskuni  8421  gruina  8456  mreexexd  13566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-card 7588
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