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Theorem alephiso 7609
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 7576 . . . . . 6  |-  aleph  Fn  On
2 isinfcard 7603 . . . . . . . 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 2360 . . . . . 6  |-  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
5 df-fo 4606 . . . . . 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 5308 . . . . 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 7595 . . . . . 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 2571 . . . 4  |-  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z )
12 dff13 5635 . . . 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 4607 . . 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 7587 . . . 4  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  e.  z  <-> 
( aleph `  y )  e.  ( aleph `  z )
) )
17 epel 4201 . . . 4  |-  ( y  _E  z  <->  y  e.  z )
18 fvex 5391 . . . . 5  |-  ( aleph `  z )  e.  _V
1918epelc 4200 . . . 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 2571 . 2  |-  A. y  e.  On  A. z  e.  On  ( y  _E  z  <->  ( aleph `  y
)  _E  ( aleph `  z ) )
22 df-isom 4609 . 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 2239   A.wral 2509    C_ wss 3078   class class class wbr 3920    _E cep 4196   Oncon0 4285   omcom 4547   ran crn 4581    Fn wfn 4587   -->wf 4588   -1-1->wf1 4589   -onto->wfo 4590   -1-1-onto->wf1o 4591   ` cfv 4592    Isom wiso 4593   cardccrd 7452   alephcale 7453
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-card 7456  df-aleph 7457
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