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Theorem cardid2 7581
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardid2  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
Dummy variable  y is distinct from all other variables.

Proof of Theorem cardid2
StepHypRef Expression
1 cardval3 7580 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
2 ssrab2 3259 . . . 4  |-  { y  e.  On  |  y 
~~  A }  C_  On
3 fvex 5499 . . . . . 6  |-  ( card `  A )  e.  _V
41, 3syl6eqelr 2373 . . . . 5  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )
5 intex 4170 . . . . 5  |-  ( { y  e.  On  | 
y  ~~  A }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  A }  e.  _V )
64, 5sylibr 205 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =/=  (/) )
7 onint 4585 . . . 4  |-  ( ( { y  e.  On  |  y  ~~  A }  C_  On  /\  { y  e.  On  |  y 
~~  A }  =/=  (/) )  ->  |^| { y  e.  On  |  y 
~~  A }  e.  { y  e.  On  | 
y  ~~  A }
)
82, 6, 7sylancr 646 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  { y  e.  On  |  y  ~~  A }
)
91, 8eqeltrd 2358 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  e.  { y  e.  On  |  y  ~~  A }
)
10 breq1 4027 . . . 4  |-  ( y  =  ( card `  A
)  ->  ( y  ~~  A  <->  ( card `  A
)  ~~  A )
)
1110elrab 2924 . . 3  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  <->  ( ( card `  A
)  e.  On  /\  ( card `  A )  ~~  A ) )
1211simprbi 452 . 2  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  ->  ( card `  A
)  ~~  A )
139, 12syl 17 1  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1685    =/= wne 2447   {crab 2548   _Vcvv 2789    C_ wss 3153   (/)c0 3456   |^|cint 3863   class class class wbr 4024   Oncon0 4391   dom cdm 4688   ` cfv 5221    ~~ cen 6855   cardccrd 7563
This theorem is referenced by:  isnum3  7582  oncardid  7584  cardidm  7587  ficardom  7589  ficardid  7590  cardnn  7591  cardnueq0  7592  carden2a  7594  carden2b  7595  carddomi2  7598  sdomsdomcardi  7599  cardsdomelir  7601  cardsdomel  7602  infxpidm2  7639  dfac8b  7653  numdom  7660  alephnbtwn2  7694  alephsucdom  7701  infenaleph  7713  dfac12r  7767  cardacda  7819  pwsdompw  7825  cff1  7879  cfflb  7880  cflim2  7884  cfss  7886  cfslb  7887  domtriomlem  8063  cardid  8164  cardidg  8165  carden  8168  sdomsdomcard  8177  hargch  8294  gch2  8296  tskcard  8398  tskuni  8400  hashkf  11333
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-en 6859  df-card 7567
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