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Theorem cardid2 7540
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 )

Proof of Theorem cardid2
StepHypRef Expression
1 cardval3 7539 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
2 ssrab2 3219 . . . 4  |-  { y  e.  On  |  y 
~~  A }  C_  On
3 fvex 5458 . . . . . 6  |-  ( card `  A )  e.  _V
41, 3syl6eqelr 2345 . . . . 5  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )
5 intex 4129 . . . . 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 4544 . . . 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 647 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  { y  e.  On  |  y  ~~  A }
)
91, 8eqeltrd 2330 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  e.  { y  e.  On  |  y  ~~  A }
)
10 breq1 3986 . . . 4  |-  ( y  =  ( card `  A
)  ->  ( y  ~~  A  <->  ( card `  A
)  ~~  A )
)
1110elrab 2891 . . 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 1621    =/= wne 2419   {crab 2520   _Vcvv 2757    C_ wss 3113   (/)c0 3416   |^|cint 3822   class class class wbr 3983   Oncon0 4350   dom cdm 4647   ` cfv 4659    ~~ cen 6814   cardccrd 7522
This theorem is referenced by:  isnum3  7541  oncardid  7543  cardidm  7546  ficardom  7548  ficardid  7549  cardnn  7550  cardnueq0  7551  carden2a  7553  carden2b  7554  carddomi2  7557  sdomsdomcardi  7558  cardsdomelir  7560  cardsdomel  7561  infxpidm2  7598  dfac8b  7612  numdom  7619  alephnbtwn2  7653  alephsucdom  7660  infenaleph  7672  dfac12r  7726  cardacda  7778  pwsdompw  7784  cff1  7838  cfflb  7839  cflim2  7843  cfss  7845  cfslb  7846  domtriomlem  8022  cardid  8123  cardidg  8124  carden  8127  sdomsdomcard  8136  hargch  8253  gch2  8255  tskcard  8357  tskuni  8359  hashkf  11291
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-en 6818  df-card 7526
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