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Theorem alephon 7696
Description: An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
alephon  |-  ( aleph `  A )  e.  On

Proof of Theorem alephon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfnon 7692 . . 3  |-  aleph  Fn  On
2 fveq2 5525 . . . . . 6  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32eleq1d 2349 . . . . 5  |-  ( x  =  (/)  ->  ( (
aleph `  x )  e.  On  <->  ( aleph `  (/) )  e.  On ) )
4 fveq2 5525 . . . . . 6  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54eleq1d 2349 . . . . 5  |-  ( x  =  y  ->  (
( aleph `  x )  e.  On  <->  ( aleph `  y
)  e.  On ) )
6 fveq2 5525 . . . . . 6  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
76eleq1d 2349 . . . . 5  |-  ( x  =  suc  y  -> 
( ( aleph `  x
)  e.  On  <->  ( aleph ` 
suc  y )  e.  On ) )
8 aleph0 7693 . . . . . 6  |-  ( aleph `  (/) )  =  om
9 omelon 7347 . . . . . 6  |-  om  e.  On
108, 9eqeltri 2353 . . . . 5  |-  ( aleph `  (/) )  e.  On
11 alephsuc 7695 . . . . . . 7  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
12 harcl 7275 . . . . . . 7  |-  (har `  ( aleph `  y )
)  e.  On
1311, 12syl6eqel 2371 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  e.  On )
1413a1d 22 . . . . 5  |-  ( y  e.  On  ->  (
( aleph `  y )  e.  On  ->  ( aleph ` 
suc  y )  e.  On ) )
15 vex 2791 . . . . . . 7  |-  x  e. 
_V
16 fvex 5539 . . . . . . 7  |-  ( aleph `  y )  e.  _V
1715, 16iunonOLD 6356 . . . . . 6  |-  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  U_ y  e.  x  ( aleph `  y
)  e.  On )
18 alephlim 7694 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
1915, 18mpan 651 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2019eleq1d 2349 . . . . . 6  |-  ( Lim  x  ->  ( ( aleph `  x )  e.  On  <->  U_ y  e.  x  ( aleph `  y )  e.  On ) )
2117, 20syl5ibr 212 . . . . 5  |-  ( Lim  x  ->  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  ( aleph `  x )  e.  On ) )
223, 5, 7, 5, 10, 14, 21tfinds 4650 . . . 4  |-  ( y  e.  On  ->  ( aleph `  y )  e.  On )
2322rgen 2608 . . 3  |-  A. y  e.  On  ( aleph `  y
)  e.  On
24 ffnfv 5685 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. y  e.  On  ( aleph `  y
)  e.  On ) )
251, 23, 24mpbir2an 886 . 2  |-  aleph : On --> On
26 0elon 4445 . 2  |-  (/)  e.  On
2725, 26f0cli 5671 1  |-  ( aleph `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    = 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    Fn wfn 5250   -->wf 5251   ` cfv 5255  harchar 7270   alephcale 7569
This theorem is referenced by:  alephnbtwn  7698  alephnbtwn2  7699  alephordilem1  7700  alephord  7702  alephord2  7703  alephord3  7705  alephsucdom  7706  alephsuc2  7707  alephf1  7712  alephsdom  7713  alephdom2  7714  alephle  7715  cardaleph  7716  alephf1ALT  7730  alephfp  7735  dfac12k  7773  alephsing  7902  alephval2  8194  alephadd  8199  alephmul  8200  alephexp1  8201  alephsuc3  8202  alephreg  8204  pwcfsdom  8205  cfpwsdom  8206  gchaleph  8297  gchaleph2  8298  gch2  8301
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-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-en 6864  df-dom 6865  df-oi 7225  df-har 7272  df-aleph 7573
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