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Theorem cardid 8348
Description: Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
cardval.1  |-  A  e. 
_V
Assertion
Ref Expression
cardid  |-  ( card `  A )  ~~  A

Proof of Theorem cardid
StepHypRef Expression
1 cardval.1 . 2  |-  A  e. 
_V
2 numth3 8276 . 2  |-  ( A  e.  _V  ->  A  e.  dom  card )
3 cardid2 7766 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
41, 2, 3mp2b 10 1  |-  ( card `  A )  ~~  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   _Vcvv 2892   class class class wbr 4146   dom cdm 4811   ` cfv 5387    ~~ cen 7035   cardccrd 7748
This theorem is referenced by:  unsnen  8354  alephval2  8373  cfpwsdom  8385  inar1  8576  gruina  8619
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-ac2 8269
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-riota 6478  df-recs 6562  df-en 7039  df-card 7752  df-ac 7923
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