MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardon Unicode version

Theorem cardon 7791
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 7790 . 2  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 0elon 4598 . 2  |-  (/)  e.  On
31, 2f0cli 5843 1  |-  ( card `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   {cab 2394   E.wrex 2671   class class class wbr 4176   Oncon0 4545   ` cfv 5417    ~~ cen 7069   cardccrd 7782
This theorem is referenced by:  isnum3  7801  cardidm  7806  ficardom  7808  cardne  7812  carden2b  7814  cardlim  7819  cardsdomelir  7820  cardsdomel  7821  iscard  7822  iscard2  7823  carddom2  7824  carduni  7828  cardom  7833  cardsdom2  7835  domtri2  7836  cardval2  7838  infxpidm2  7858  dfac8b  7872  numdom  7879  indcardi  7882  alephnbtwn  7912  alephnbtwn2  7913  alephsucdom  7920  cardaleph  7930  iscard3  7934  alephinit  7936  alephsson  7941  alephval3  7951  dfac12r  7986  dfac12k  7987  cardacda  8038  cdanum  8039  pwsdompw  8044  cff  8088  cardcf  8092  cfon  8095  cfeq0  8096  cfsuc  8097  cff1  8098  cfflb  8099  cflim2  8103  cfss  8105  fin1a2lem9  8248  ttukeylem6  8354  ttukeylem7  8355  unsnen  8388  inar1  8610  tskcard  8616  tskuni  8618  gruina  8653  mreexexd  13832
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-card 7786
  Copyright terms: Public domain W3C validator