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Theorem alephiso 8267
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
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfnon 8234 . . . . . 6  |-  aleph  Fn  On
2 isinfcard 8261 . . . . . . . 8  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  <->  x  e.  ran  aleph )
32bicomi 202 . . . . . . 7  |-  ( x  e.  ran  aleph  <->  ( om  C_  x  /\  ( card `  x )  =  x ) )
43abbi2i 2553 . . . . . 6  |-  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
5 df-fo 5423 . . . . . 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 911 . . . . 5  |-  aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
7 fof 5619 . . . . 5  |-  ( aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  ->  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x
)  =  x ) } )
86, 7ax-mp 5 . . . 4  |-  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
9 aleph11 8253 . . . . . 6  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  <->  y  =  z ) )
109biimpd 207 . . . . 5  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  ->  y  =  z ) )
1110rgen2a 2781 . . . 4  |-  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z )
12 dff13 5970 . . . 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 911 . . 3  |-  aleph : On -1-1-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
14 df-f1o 5424 . . 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 911 . 2  |-  aleph : On -1-1-onto-> {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
16 alephord2 8245 . . . 4  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  e.  z  <-> 
( aleph `  y )  e.  ( aleph `  z )
) )
17 epel 4634 . . . 4  |-  ( y  _E  z  <->  y  e.  z )
18 fvex 5700 . . . . 5  |-  ( aleph `  z )  e.  _V
1918epelc 4633 . . . 4  |-  ( (
aleph `  y )  _E  ( aleph `  z )  <->  (
aleph `  y )  e.  ( aleph `  z )
)
2016, 17, 193bitr4g 288 . . 3  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  _E  z  <->  (
aleph `  y )  _E  ( aleph `  z )
) )
2120rgen2a 2781 . 2  |-  A. y  e.  On  A. z  e.  On  ( y  _E  z  <->  ( aleph `  y
)  _E  ( aleph `  z ) )
22 df-isom 5426 . 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 911 1  |-  aleph  Isom  _E  ,  _E  ( On ,  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2428   A.wral 2714    C_ wss 3327   class class class wbr 4291    _E cep 4629   Oncon0 4718   ran crn 4840    Fn wfn 5412   -->wf 5413   -1-1->wf1 5414   -onto->wfo 5415   -1-1-onto->wf1o 5416   ` cfv 5417    Isom wiso 5418   omcom 6475   cardccrd 8104   alephcale 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-om 6476  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-oi 7723  df-har 7772  df-card 8108  df-aleph 8109
This theorem is referenced by: (None)
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