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Theorem alephcard 7877
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 5661 . . . . 5  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
21fveq2d 5665 . . . 4  |-  ( x  =  (/)  ->  ( card `  ( aleph `  x )
)  =  ( card `  ( aleph `  (/) ) ) )
32, 1eqeq12d 2394 . . 3  |-  ( x  =  (/)  ->  ( (
card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) ) ) )
4 fveq2 5661 . . . . 5  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54fveq2d 5665 . . . 4  |-  ( x  =  y  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  y
) ) )
65, 4eqeq12d 2394 . . 3  |-  ( x  =  y  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  y )
)  =  ( aleph `  y ) ) )
7 fveq2 5661 . . . . 5  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
87fveq2d 5665 . . . 4  |-  ( x  =  suc  y  -> 
( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  suc  y ) ) )
98, 7eqeq12d 2394 . . 3  |-  ( x  =  suc  y  -> 
( ( card `  ( aleph `  x ) )  =  ( aleph `  x
)  <->  ( card `  ( aleph `  suc  y ) )  =  ( aleph ` 
suc  y ) ) )
10 fveq2 5661 . . . . 5  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
1110fveq2d 5665 . . . 4  |-  ( x  =  A  ->  ( card `  ( aleph `  x
) )  =  (
card `  ( aleph `  A
) ) )
1211, 10eqeq12d 2394 . . 3  |-  ( x  =  A  ->  (
( card `  ( aleph `  x
) )  =  (
aleph `  x )  <->  ( card `  ( aleph `  A )
)  =  ( aleph `  A ) ) )
13 cardom 7799 . . . 4  |-  ( card `  om )  =  om
14 aleph0 7873 . . . . 5  |-  ( aleph `  (/) )  =  om
1514fveq2i 5664 . . . 4  |-  ( card `  ( aleph `  (/) ) )  =  ( card `  om )
1613, 15, 143eqtr4i 2410 . . 3  |-  ( card `  ( aleph `  (/) ) )  =  ( aleph `  (/) )
17 harcard 7791 . . . . 5  |-  ( card `  (har `  ( aleph `  y
) ) )  =  (har `  ( aleph `  y
) )
18 alephsuc 7875 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
1918fveq2d 5665 . . . . 5  |-  ( y  e.  On  ->  ( card `  ( aleph `  suc  y ) )  =  ( card `  (har `  ( aleph `  y )
) ) )
2017, 19, 183eqtr4a 2438 . . . 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 2895 . . . . . . 7  |-  x  e. 
_V
23 cardiun 7795 . . . . . . 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 7874 . . . . . . . 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 5665 . . . . 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 2422 . . . 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 4772 . 2  |-  ( A  e.  On  ->  ( card `  ( aleph `  A
) )  =  (
aleph `  A ) )
33 card0 7771 . . 3  |-  ( card `  (/) )  =  (/)
34 alephfnon 7872 . . . . . . 7  |-  aleph  Fn  On
35 fndm 5477 . . . . . . 7  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3634, 35ax-mp 8 . . . . . 6  |-  dom  aleph  =  On
3736eleq2i 2444 . . . . 5  |-  ( A  e.  dom  aleph  <->  A  e.  On )
38 ndmfv 5688 . . . . 5  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
3937, 38sylnbir 299 . . . 4  |-  ( -.  A  e.  On  ->  (
aleph `  A )  =  (/) )
4039fveq2d 5665 . . 3  |-  ( -.  A  e.  On  ->  (
card `  ( aleph `  A
) )  =  (
card `  (/) ) )
4133, 40, 393eqtr4a 2438 . 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 1717   A.wral 2642   _Vcvv 2892   (/)c0 3564   U_ciun 4028   Oncon0 4515   Lim wlim 4516   suc csuc 4517   omcom 4778   dom cdm 4811    Fn wfn 5382   ` cfv 5387  harchar 7450   cardccrd 7748   alephcale 7749
This theorem is referenced by:  alephnbtwn2  7879  alephord2  7883  alephsuc2  7887  alephislim  7890  alephsdom  7893  cardaleph  7896  cardalephex  7897  alephval3  7917  alephval2  8373  alephsuc3  8381  alephreg  8383  pwcfsdom  8384
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-oi 7405  df-har 7452  df-card 7752  df-aleph 7753
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