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Theorem alephiso 7693
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 7660 . . . . . 6  |-  aleph  Fn  On
2 isinfcard 7687 . . . . . . . 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 2369 . . . . . 6  |-  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
5 df-fo 4687 . . . . . 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 5389 . . . . 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 7679 . . . . . 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 2584 . . . 4  |-  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z )
12 dff13 5717 . . . 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 4688 . . 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 7671 . . . 4  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  e.  z  <-> 
( aleph `  y )  e.  ( aleph `  z )
) )
17 epel 4280 . . . 4  |-  ( y  _E  z  <->  y  e.  z )
18 fvex 5472 . . . . 5  |-  ( aleph `  z )  e.  _V
1918epelc 4279 . . . 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 2584 . 2  |-  A. y  e.  On  A. z  e.  On  ( y  _E  z  <->  ( aleph `  y
)  _E  ( aleph `  z ) )
22 df-isom 4690 . 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 2244   A.wral 2518    C_ wss 3127   class class class wbr 3997    _E cep 4275   Oncon0 4364   omcom 4628   ran crn 4662    Fn wfn 4668   -->wf 4669   -1-1->wf1 4670   -onto->wfo 4671   -1-1-onto->wf1o 4672   ` cfv 4673    Isom wiso 4674   cardccrd 7536   alephcale 7537
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-oi 7193  df-har 7240  df-card 7540  df-aleph 7541
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