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Theorem alephcard 7907
Description: Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephcard  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )

Proof of Theorem alephcard
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5687 . . . . 5  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
21fveq2d 5691 . . . 4  |-  ( x  =  (/)  ->  ( card `  ( aleph `  x )
)  =  ( card `  ( aleph `  (/) ) ) )
32, 1eqeq12d 2418 . . 3  |-  ( x  =  (/)  ->  ( (
card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) ) ) )
4 fveq2 5687 . . . . 5  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54fveq2d 5691 . . . 4  |-  ( x  =  y  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  y
) ) )
65, 4eqeq12d 2418 . . 3  |-  ( x  =  y  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  y )
)  =  ( aleph `  y ) ) )
7 fveq2 5687 . . . . 5  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
87fveq2d 5691 . . . 4  |-  ( x  =  suc  y  -> 
( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  suc  y ) ) )
98, 7eqeq12d 2418 . . 3  |-  ( x  =  suc  y  -> 
( ( card `  ( aleph `  x ) )  =  ( aleph `  x
)  <->  ( card `  ( aleph `  suc  y ) )  =  ( aleph ` 
suc  y ) ) )
10 fveq2 5687 . . . . 5  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
1110fveq2d 5691 . . . 4  |-  ( x  =  A  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  A
) ) )
1211, 10eqeq12d 2418 . . 3  |-  ( x  =  A  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  A )
)  =  ( aleph `  A ) ) )
13 cardom 7829 . . . 4  |-  ( card `  om )  =  om
14 aleph0 7903 . . . . 5  |-  ( aleph `  (/) )  =  om
1514fveq2i 5690 . . . 4  |-  ( card `  ( aleph `  (/) ) )  =  ( card `  om )
1613, 15, 143eqtr4i 2434 . . 3  |-  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) )
17 harcard 7821 . . . . 5  |-  ( card `  (har `  ( aleph `  y
) ) )  =  (har `  ( aleph `  y
) )
18 alephsuc 7905 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
1918fveq2d 5691 . . . . 5  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( card `  (har `  ( aleph `  y )
) ) )
2017, 19, 183eqtr4a 2462 . . . 4  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) )
2120a1d 23 . . 3  |-  ( y  e.  On  ->  (
( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) ) )
22 vex 2919 . . . . . . 7  |-  x  e. 
_V
23 cardiun 7825 . . . . . . 7  |-  ( x  e.  _V  ->  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  U_ y  e.  x  ( aleph `  y
) )  =  U_ y  e.  x  ( aleph `  y ) ) )
2422, 23ax-mp 8 . . . . . 6  |-  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  U_ y  e.  x  ( aleph `  y
) )  =  U_ y  e.  x  ( aleph `  y ) )
2524adantl 453 . . . . 5  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  U_ y  e.  x  ( aleph `  y ) )  = 
U_ y  e.  x  ( aleph `  y )
)
26 alephlim 7904 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
2722, 26mpan 652 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2827adantr 452 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( aleph `  x
)  =  U_ y  e.  x  ( aleph `  y ) )
2928fveq2d 5691 . . . . 5  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  ( aleph `  x ) )  =  ( card `  U_ y  e.  x  ( aleph `  y ) ) )
3025, 29, 283eqtr4d 2446 . . . 4  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  ( aleph `  x ) )  =  ( aleph `  x
) )
3130ex 424 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  x
) )  =  (
aleph `  x ) ) )
323, 6, 9, 12, 16, 21, 31tfinds 4798 . 2  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
33 card0 7801 . . 3  |-  ( card `  (/) )  =  (/)
34 alephfnon 7902 . . . . . . 7  |-  aleph  Fn  On
35 fndm 5503 . . . . . . 7  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3634, 35ax-mp 8 . . . . . 6  |-  dom  aleph  =  On
3736eleq2i 2468 . . . . 5  |-  ( A  e.  dom  aleph  <->  A  e.  On )
38 ndmfv 5714 . . . . 5  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
3937, 38sylnbir 299 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  A )  =  (/) )
4039fveq2d 5691 . . 3  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
card `  (/) ) )
4133, 40, 393eqtr4a 2462 . 2  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
aleph `  A ) )
4232, 41pm2.61i 158 1  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   (/)c0 3588   U_ciun 4053   Oncon0 4541   Lim wlim 4542   suc csuc 4543   omcom 4804   dom cdm 4837    Fn wfn 5408   ` cfv 5413  harchar 7480   cardccrd 7778   alephcale 7779
This theorem is referenced by:  alephnbtwn2  7909  alephord2  7913  alephsuc2  7917  alephislim  7920  alephsdom  7923  cardaleph  7926  cardalephex  7927  alephval3  7947  alephval2  8403  alephsuc3  8411  alephreg  8413  pwcfsdom  8414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783
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