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Theorem alephiso 7658
Description: Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)
Assertion
Ref Expression
alephiso  |-  aleph  Isom  _E  ,  _E  ( On ,  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )

Proof of Theorem alephiso
StepHypRef Expression
1 alephfnon 7625 . . . . . 6  |-  aleph  Fn  On
2 isinfcard 7652 . . . . . . . 8  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  <->  x  e.  ran  aleph )
32bicomi 195 . . . . . . 7  |-  ( x  e.  ran  aleph  <->  ( om  C_  x  /\  ( card `  x )  =  x ) )
43abbi2i 2367 . . . . . 6  |-  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
5 df-fo 4652 . . . . . 6  |-  ( aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  <->  ( aleph  Fn  On  /\  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } ) )
61, 4, 5mpbir2an 891 . . . . 5  |-  aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
7 fof 5354 . . . . 5  |-  ( aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  ->  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x
)  =  x ) } )
86, 7ax-mp 10 . . . 4  |-  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
9 aleph11 7644 . . . . . 6  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  <->  y  =  z ) )
109biimpd 200 . . . . 5  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  ->  y  =  z ) )
1110rgen2a 2580 . . . 4  |-  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z )
12 dff13 5682 . . . 4  |-  ( aleph : On -1-1-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  <->  ( aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x
)  =  x ) }  /\  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z ) ) )
138, 11, 12mpbir2an 891 . . 3  |-  aleph : On -1-1-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
14 df-f1o 4653 . . 3  |-  ( aleph : On -1-1-onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  <->  ( aleph : On -1-1-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  /\  aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } ) )
1513, 6, 14mpbir2an 891 . 2  |-  aleph : On -1-1-onto-> {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
16 alephord2 7636 . . . 4  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  e.  z  <-> 
( aleph `  y )  e.  ( aleph `  z )
) )
17 epel 4245 . . . 4  |-  ( y  _E  z  <->  y  e.  z )
18 fvex 5437 . . . . 5  |-  ( aleph `  z )  e.  _V
1918epelc 4244 . . . 4  |-  ( (
aleph `  y )  _E  ( aleph `  z )  <->  (
aleph `  y )  e.  ( aleph `  z )
)
2016, 17, 193bitr4g 281 . . 3  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  _E  z  <->  (
aleph `  y )  _E  ( aleph `  z )
) )
2120rgen2a 2580 . 2  |-  A. y  e.  On  A. z  e.  On  ( y  _E  z  <->  ( aleph `  y
)  _E  ( aleph `  z ) )
22 df-isom 4655 . 2  |-  ( aleph  Isom 
_E  ,  _E  ( On ,  { x  |  ( om  C_  x  /\  ( card `  x
)  =  x ) } )  <->  ( aleph : On -1-1-onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  /\  A. y  e.  On  A. z  e.  On  (
y  _E  z  <->  ( aleph `  y )  _E  ( aleph `  z ) ) ) )
2315, 21, 22mpbir2an 891 1  |-  aleph  Isom  _E  ,  _E  ( On ,  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516    C_ wss 3094   class class class wbr 3963    _E cep 4240   Oncon0 4329   omcom 4593   ran crn 4627    Fn wfn 4633   -->wf 4634   -1-1->wf1 4635   -onto->wfo 4636   -1-1-onto->wf1o 4637   ` cfv 4638    Isom wiso 4639   cardccrd 7501   alephcale 7502
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-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506
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