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Theorem alephcard 8526
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 5887 . . . . 5  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
21fveq2d 5891 . . . 4  |-  ( x  =  (/)  ->  ( card `  ( aleph `  x )
)  =  ( card `  ( aleph `  (/) ) ) )
32, 1eqeq12d 2476 . . 3  |-  ( x  =  (/)  ->  ( (
card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) ) ) )
4 fveq2 5887 . . . . 5  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54fveq2d 5891 . . . 4  |-  ( x  =  y  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  y
) ) )
65, 4eqeq12d 2476 . . 3  |-  ( x  =  y  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  y )
)  =  ( aleph `  y ) ) )
7 fveq2 5887 . . . . 5  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
87fveq2d 5891 . . . 4  |-  ( x  =  suc  y  -> 
( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  suc  y ) ) )
98, 7eqeq12d 2476 . . 3  |-  ( x  =  suc  y  -> 
( ( card `  ( aleph `  x ) )  =  ( aleph `  x
)  <->  ( card `  ( aleph `  suc  y ) )  =  ( aleph ` 
suc  y ) ) )
10 fveq2 5887 . . . . 5  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
1110fveq2d 5891 . . . 4  |-  ( x  =  A  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  A
) ) )
1211, 10eqeq12d 2476 . . 3  |-  ( x  =  A  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  A )
)  =  ( aleph `  A ) ) )
13 cardom 8445 . . . 4  |-  ( card `  om )  =  om
14 aleph0 8522 . . . . 5  |-  ( aleph `  (/) )  =  om
1514fveq2i 5890 . . . 4  |-  ( card `  ( aleph `  (/) ) )  =  ( card `  om )
1613, 15, 143eqtr4i 2493 . . 3  |-  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) )
17 harcard 8437 . . . . 5  |-  ( card `  (har `  ( aleph `  y
) ) )  =  (har `  ( aleph `  y
) )
18 alephsuc 8524 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
1918fveq2d 5891 . . . . 5  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( card `  (har `  ( aleph `  y )
) ) )
2017, 19, 183eqtr4a 2521 . . . 4  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) )
2120a1d 26 . . 3  |-  ( y  e.  On  ->  (
( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) ) )
22 vex 3059 . . . . . . 7  |-  x  e. 
_V
23 cardiun 8441 . . . . . . 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 5 . . . . . 6  |-  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  U_ y  e.  x  ( aleph `  y
) )  =  U_ y  e.  x  ( aleph `  y ) )
2524adantl 472 . . . . 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 8523 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
2722, 26mpan 681 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2827adantr 471 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( aleph `  x
)  =  U_ y  e.  x  ( aleph `  y ) )
2928fveq2d 5891 . . . . 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 2505 . . . 4  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  ( aleph `  x ) )  =  ( aleph `  x
) )
3130ex 440 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  x
) )  =  (
aleph `  x ) ) )
323, 6, 9, 12, 16, 21, 31tfinds 6712 . 2  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
33 card0 8417 . . 3  |-  ( card `  (/) )  =  (/)
34 alephfnon 8521 . . . . . . 7  |-  aleph  Fn  On
35 fndm 5696 . . . . . . 7  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3634, 35ax-mp 5 . . . . . 6  |-  dom  aleph  =  On
3736eleq2i 2531 . . . . 5  |-  ( A  e.  dom  aleph  <->  A  e.  On )
38 ndmfv 5911 . . . . 5  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
3937, 38sylnbir 313 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  A )  =  (/) )
4039fveq2d 5891 . . 3  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
card `  (/) ) )
4133, 40, 393eqtr4a 2521 . 2  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
aleph `  A ) )
4232, 41pm2.61i 169 1  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   _Vcvv 3056   (/)c0 3742   U_ciun 4291   dom cdm 4852   Oncon0 5441   Lim wlim 5442   suc csuc 5443    Fn wfn 5595   ` cfv 5600   omcom 6718  harchar 8096   cardccrd 8394   alephcale 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-oi 8050  df-har 8098  df-card 8398  df-aleph 8399
This theorem is referenced by:  alephnbtwn2  8528  alephord2  8532  alephsuc2  8536  alephislim  8539  alephsdom  8542  cardaleph  8545  cardalephex  8546  alephval3  8566  alephval2  9022  alephsuc3  9030  alephreg  9032  pwcfsdom  9033
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