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Theorem alephiso 7741
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 7708 . . . . . 6  |-  aleph  Fn  On
2 isinfcard 7735 . . . . . . . 8  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  <->  x  e.  ran  aleph )
32bicomi 193 . . . . . . 7  |-  ( x  e.  ran  aleph  <->  ( om  C_  x  /\  ( card `  x )  =  x ) )
43abbi2i 2407 . . . . . 6  |-  ran  aleph  =  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
5 df-fo 5277 . . . . . 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 886 . . . . 5  |-  aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
7 fof 5467 . . . . 5  |-  ( aleph : On -onto-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }  ->  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x
)  =  x ) } )
86, 7ax-mp 8 . . . 4  |-  aleph : On --> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
9 aleph11 7727 . . . . . 6  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  <->  y  =  z ) )
109biimpd 198 . . . . 5  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( ( aleph `  y
)  =  ( aleph `  z )  ->  y  =  z ) )
1110rgen2a 2622 . . . 4  |-  A. y  e.  On  A. z  e.  On  ( ( aleph `  y )  =  (
aleph `  z )  -> 
y  =  z )
12 dff13 5799 . . . 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 886 . . 3  |-  aleph : On -1-1-> { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
14 df-f1o 5278 . . 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 886 . 2  |-  aleph : On -1-1-onto-> {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) }
16 alephord2 7719 . . . 4  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  e.  z  <-> 
( aleph `  y )  e.  ( aleph `  z )
) )
17 epel 4324 . . . 4  |-  ( y  _E  z  <->  y  e.  z )
18 fvex 5555 . . . . 5  |-  ( aleph `  z )  e.  _V
1918epelc 4323 . . . 4  |-  ( (
aleph `  y )  _E  ( aleph `  z )  <->  (
aleph `  y )  e.  ( aleph `  z )
)
2016, 17, 193bitr4g 279 . . 3  |-  ( ( y  e.  On  /\  z  e.  On )  ->  ( y  _E  z  <->  (
aleph `  y )  _E  ( aleph `  z )
) )
2120rgen2a 2622 . 2  |-  A. y  e.  On  A. z  e.  On  ( y  _E  z  <->  ( aleph `  y
)  _E  ( aleph `  z ) )
22 df-isom 5280 . 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 886 1  |-  aleph  Isom  _E  ,  _E  ( On ,  {
x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165   class class class wbr 4039    _E cep 4319   Oncon0 4408   omcom 4672   ran crn 4706    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   cardccrd 7584   alephcale 7585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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