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Theorem cardon 8228
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 8227 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4883 . 2  |-  (/)  e.  On
31, 2f0cli 5966 1  |-  ( card `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   {cab 2439   E.wrex 2800   class class class wbr 4403   Oncon0 4830   ` cfv 5529    ~~ cen 7420   cardccrd 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-card 8223
This theorem is referenced by:  isnum3  8238  cardidm  8243  ficardom  8245  cardne  8249  carden2b  8251  cardlim  8256  cardsdomelir  8257  cardsdomel  8258  iscard  8259  iscard2  8260  carddom2  8261  carduni  8265  cardom  8270  cardsdom2  8272  domtri2  8273  cardval2  8275  infxpidm2  8297  dfac8b  8315  numdom  8322  indcardi  8325  alephnbtwn  8355  alephnbtwn2  8356  alephsucdom  8363  cardaleph  8373  iscard3  8377  alephinit  8379  alephsson  8384  alephval3  8394  dfac12r  8429  dfac12k  8430  cardacda  8481  cdanum  8482  pwsdompw  8487  cff  8531  cardcf  8535  cfon  8538  cfeq0  8539  cfsuc  8540  cff1  8541  cfflb  8542  cflim2  8546  cfss  8548  fin1a2lem9  8691  ttukeylem6  8797  ttukeylem7  8798  unsnen  8831  inar1  9056  tskcard  9062  tskuni  9064  gruina  9099  mreexexd  14708
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