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Theorem cardid2 8351
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardval3 8350 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
2 ssrab2 3581 . . . 4  |-  { y  e.  On  |  y 
~~  A }  C_  On
3 fvex 5882 . . . . . 6  |-  ( card `  A )  e.  _V
41, 3syl6eqelr 2554 . . . . 5  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )
5 intex 4612 . . . . 5  |-  ( { y  e.  On  | 
y  ~~  A }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  A }  e.  _V )
64, 5sylibr 212 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =/=  (/) )
7 onint 6629 . . . 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 663 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  { y  e.  On  |  y  ~~  A }
)
91, 8eqeltrd 2545 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  e.  { y  e.  On  |  y  ~~  A }
)
10 breq1 4459 . . . 4  |-  ( y  =  ( card `  A
)  ->  ( y  ~~  A  <->  ( card `  A
)  ~~  A )
)
1110elrab 3257 . . 3  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  <->  ( ( card `  A
)  e.  On  /\  ( card `  A )  ~~  A ) )
1211simprbi 464 . 2  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  ->  ( card `  A
)  ~~  A )
139, 12syl 16 1  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   |^|cint 4288   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~~ cen 7532   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-en 7536  df-card 8337
This theorem is referenced by:  isnum3  8352  oncardid  8354  cardidm  8357  ficardom  8359  ficardid  8360  cardnn  8361  cardnueq0  8362  carden2a  8364  carden2b  8365  carddomi2  8368  sdomsdomcardi  8369  cardsdomelir  8371  cardsdomel  8372  infxpidm2  8411  dfac8b  8429  numdom  8436  alephnbtwn2  8470  alephsucdom  8477  infenaleph  8489  dfac12r  8543  cardacda  8595  pwsdompw  8601  cff1  8655  cfflb  8656  cflim2  8660  cfss  8662  cfslb  8663  domtriomlem  8839  cardid  8939  cardidg  8940  carden  8943  sdomsdomcard  8952  hargch  9068  gch2  9070  hashkf  12410
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