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Theorem alephcard 7665
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
StepHypRef Expression
1 fveq2 5458 . . . . 5  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
21fveq2d 5462 . . . 4  |-  ( x  =  (/)  ->  ( card `  ( aleph `  x )
)  =  ( card `  ( aleph `  (/) ) ) )
32, 1eqeq12d 2272 . . 3  |-  ( x  =  (/)  ->  ( (
card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) ) ) )
4 fveq2 5458 . . . . 5  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54fveq2d 5462 . . . 4  |-  ( x  =  y  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  y
) ) )
65, 4eqeq12d 2272 . . 3  |-  ( x  =  y  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  y )
)  =  ( aleph `  y ) ) )
7 fveq2 5458 . . . . 5  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
87fveq2d 5462 . . . 4  |-  ( x  =  suc  y  -> 
( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  suc  y ) ) )
98, 7eqeq12d 2272 . . 3  |-  ( x  =  suc  y  -> 
( ( card `  ( aleph `  x ) )  =  ( aleph `  x
)  <->  ( card `  ( aleph `  suc  y ) )  =  ( aleph ` 
suc  y ) ) )
10 fveq2 5458 . . . . 5  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
1110fveq2d 5462 . . . 4  |-  ( x  =  A  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  A
) ) )
1211, 10eqeq12d 2272 . . 3  |-  ( x  =  A  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  A )
)  =  ( aleph `  A ) ) )
13 cardom 7587 . . . 4  |-  ( card `  om )  =  om
14 aleph0 7661 . . . . 5  |-  ( aleph `  (/) )  =  om
1514fveq2i 5461 . . . 4  |-  ( card `  ( aleph `  (/) ) )  =  ( card `  om )
1613, 15, 143eqtr4i 2288 . . 3  |-  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) )
17 harcard 7579 . . . . 5  |-  ( card `  (har `  ( aleph `  y
) ) )  =  (har `  ( aleph `  y
) )
18 alephsuc 7663 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
1918fveq2d 5462 . . . . 5  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( card `  (har `  ( aleph `  y )
) ) )
2017, 19, 183eqtr4a 2316 . . . 4  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) )
2120a1d 24 . . 3  |-  ( y  e.  On  ->  (
( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  suc  y ) )  =  ( aleph `  suc  y ) ) )
22 vex 2766 . . . . . . 7  |-  x  e. 
_V
23 cardiun 7583 . . . . . . 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 10 . . . . . 6  |-  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  U_ y  e.  x  ( aleph `  y
) )  =  U_ y  e.  x  ( aleph `  y ) )
2524adantl 454 . . . . 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 7662 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
2722, 26mpan 654 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2827adantr 453 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( aleph `  x
)  =  U_ y  e.  x  ( aleph `  y ) )
2928fveq2d 5462 . . . . 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 2300 . . . 4  |-  ( ( Lim  x  /\  A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y ) )  ->  ( card `  ( aleph `  x ) )  =  ( aleph `  x
) )
3130ex 425 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ( card `  ( aleph `  y
) )  =  (
aleph `  y )  -> 
( card `  ( aleph `  x
) )  =  (
aleph `  x ) ) )
323, 6, 9, 12, 16, 21, 31tfinds 4622 . 2  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
33 card0 7559 . . 3  |-  ( card `  (/) )  =  (/)
34 alephfnon 7660 . . . . . . 7  |-  aleph  Fn  On
35 fndm 5281 . . . . . . 7  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3634, 35ax-mp 10 . . . . . 6  |-  dom  aleph  =  On
3736eleq2i 2322 . . . . 5  |-  ( A  e.  dom  aleph  <->  A  e.  On )
38 ndmfv 5486 . . . . 5  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
3937, 38sylnbir 300 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  A )  =  (/) )
4039fveq2d 5462 . . 3  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
card `  (/) ) )
4133, 40, 393eqtr4a 2316 . 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 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763   (/)c0 3430   U_ciun 3879   Oncon0 4364   Lim wlim 4365   suc csuc 4366   omcom 4628   dom cdm 4661    Fn wfn 4668   ` cfv 4673  harchar 7238   cardccrd 7536   alephcale 7537
This theorem is referenced by:  alephnbtwn2  7667  alephord2  7671  alephsuc2  7675  alephislim  7678  alephsdom  7681  cardaleph  7684  cardalephex  7685  alephval3  7705  alephval2  8162  alephsuc3  8170  alephreg  8172  pwcfsdom  8173
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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