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

Theorem cardnn 7686
Description: The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardnn  |-  ( A  e.  om  ->  ( card `  A )  =  A )

Proof of Theorem cardnn
StepHypRef Expression
1 nnon 4744 . . 3  |-  ( A  e.  om  ->  A  e.  On )
2 onenon 7672 . . 3  |-  ( A  e.  On  ->  A  e.  dom  card )
3 cardid2 7676 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
41, 2, 33syl 18 . 2  |-  ( A  e.  om  ->  ( card `  A )  ~~  A )
5 nnfi 7141 . . . 4  |-  ( A  e.  om  ->  A  e.  Fin )
6 ficardom 7684 . . . 4  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
75, 6syl 15 . . 3  |-  ( A  e.  om  ->  ( card `  A )  e. 
om )
8 nneneq 7132 . . 3  |-  ( ( ( card `  A
)  e.  om  /\  A  e.  om )  ->  ( ( card `  A
)  ~~  A  <->  ( card `  A )  =  A ) )
97, 8mpancom 650 . 2  |-  ( A  e.  om  ->  (
( card `  A )  ~~  A  <->  ( card `  A
)  =  A ) )
104, 9mpbid 201 1  |-  ( A  e.  om  ->  ( card `  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1642    e. wcel 1710   class class class wbr 4104   Oncon0 4474   omcom 4738   dom cdm 4771   ` cfv 5337    ~~ cen 6948   Fincfn 6951   cardccrd 7658
This theorem is referenced by:  card1  7691  cardennn  7706  cardsucnn  7708  nnsdomel  7713  pm54.43lem  7722  iscard3  7810  nnacda  7917  ficardun  7918  ficardun2  7919  pwsdompw  7920  ackbij2  7959  sdom2en01  8018  fin23lem22  8043  fin1a2lem9  8124  ficard  8277  cfpwsdom  8296  cardfz  11124  hashgval2  11453  hashdom  11454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662
  Copyright terms: Public domain W3C validator