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Theorem alephcard 7697
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 5525 . . . . 5  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
21fveq2d 5529 . . . 4  |-  ( x  =  (/)  ->  ( card `  ( aleph `  x )
)  =  ( card `  ( aleph `  (/) ) ) )
32, 1eqeq12d 2297 . . 3  |-  ( x  =  (/)  ->  ( (
card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) ) ) )
4 fveq2 5525 . . . . 5  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54fveq2d 5529 . . . 4  |-  ( x  =  y  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  y
) ) )
65, 4eqeq12d 2297 . . 3  |-  ( x  =  y  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  y )
)  =  ( aleph `  y ) ) )
7 fveq2 5525 . . . . 5  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
87fveq2d 5529 . . . 4  |-  ( x  =  suc  y  -> 
( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  suc  y ) ) )
98, 7eqeq12d 2297 . . 3  |-  ( x  =  suc  y  -> 
( ( card `  ( aleph `  x ) )  =  ( aleph `  x
)  <->  ( card `  ( aleph `  suc  y ) )  =  ( aleph ` 
suc  y ) ) )
10 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
1110fveq2d 5529 . . . 4  |-  ( x  =  A  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  A
) ) )
1211, 10eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  A )
)  =  ( aleph `  A ) ) )
13 cardom 7619 . . . 4  |-  ( card `  om )  =  om
14 aleph0 7693 . . . . 5  |-  ( aleph `  (/) )  =  om
1514fveq2i 5528 . . . 4  |-  ( card `  ( aleph `  (/) ) )  =  ( card `  om )
1613, 15, 143eqtr4i 2313 . . 3  |-  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) )
17 harcard 7611 . . . . 5  |-  ( card `  (har `  ( aleph `  y
) ) )  =  (har `  ( aleph `  y
) )
18 alephsuc 7695 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
1918fveq2d 5529 . . . . 5  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( card `  (har `  ( aleph `  y )
) ) )
2017, 19, 183eqtr4a 2341 . . . 4  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) )
2120a1d 22 . . 3  |-  ( y  e.  On  ->  (
( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) ) )
22 vex 2791 . . . . . . 7  |-  x  e. 
_V
23 cardiun 7615 . . . . . . 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 452 . . . . 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 7694 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
2722, 26mpan 651 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2827adantr 451 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( aleph `  x
)  =  U_ y  e.  x  ( aleph `  y ) )
2928fveq2d 5529 . . . . 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 2325 . . . 4  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  ( aleph `  x ) )  =  ( aleph `  x
) )
3130ex 423 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  x
) )  =  (
aleph `  x ) ) )
323, 6, 9, 12, 16, 21, 31tfinds 4650 . 2  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
33 card0 7591 . . 3  |-  ( card `  (/) )  =  (/)
34 alephfnon 7692 . . . . . . 7  |-  aleph  Fn  On
35 fndm 5343 . . . . . . 7  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3634, 35ax-mp 8 . . . . . 6  |-  dom  aleph  =  On
3736eleq2i 2347 . . . . 5  |-  ( A  e.  dom  aleph  <->  A  e.  On )
38 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
3937, 38sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  A )  =  (/) )
4039fveq2d 5529 . . 3  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
card `  (/) ) )
4133, 40, 393eqtr4a 2341 . 2  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
aleph `  A ) )
4232, 41pm2.61i 156 1  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   (/)c0 3455   U_ciun 3905   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   ` cfv 5255  harchar 7270   cardccrd 7568   alephcale 7569
This theorem is referenced by:  alephnbtwn2  7699  alephord2  7703  alephsuc2  7707  alephislim  7710  alephsdom  7713  cardaleph  7716  cardalephex  7717  alephval3  7737  alephval2  8194  alephsuc3  8202  alephreg  8204  pwcfsdom  8205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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