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Theorem alephon 8453
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 8449 . . 3  |-  aleph  Fn  On
2 fveq2 5856 . . . . . 6  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32eleq1d 2512 . . . . 5  |-  ( x  =  (/)  ->  ( (
aleph `  x )  e.  On  <->  ( aleph `  (/) )  e.  On ) )
4 fveq2 5856 . . . . . 6  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54eleq1d 2512 . . . . 5  |-  ( x  =  y  ->  (
( aleph `  x )  e.  On  <->  ( aleph `  y
)  e.  On ) )
6 fveq2 5856 . . . . . 6  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
76eleq1d 2512 . . . . 5  |-  ( x  =  suc  y  -> 
( ( aleph `  x
)  e.  On  <->  ( aleph ` 
suc  y )  e.  On ) )
8 aleph0 8450 . . . . . 6  |-  ( aleph `  (/) )  =  om
9 omelon 8066 . . . . . 6  |-  om  e.  On
108, 9eqeltri 2527 . . . . 5  |-  ( aleph `  (/) )  e.  On
11 alephsuc 8452 . . . . . . 7  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
12 harcl 7990 . . . . . . 7  |-  (har `  ( aleph `  y )
)  e.  On
1311, 12syl6eqel 2539 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  e.  On )
1413a1d 25 . . . . 5  |-  ( y  e.  On  ->  (
( aleph `  y )  e.  On  ->  ( aleph ` 
suc  y )  e.  On ) )
15 vex 3098 . . . . . . 7  |-  x  e. 
_V
16 fvex 5866 . . . . . . 7  |-  ( aleph `  y )  e.  _V
1715, 16iunonOLD 7012 . . . . . 6  |-  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  U_ y  e.  x  ( aleph `  y
)  e.  On )
18 alephlim 8451 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
1915, 18mpan 670 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2019eleq1d 2512 . . . . . 6  |-  ( Lim  x  ->  ( ( aleph `  x )  e.  On  <->  U_ y  e.  x  ( aleph `  y )  e.  On ) )
2117, 20syl5ibr 221 . . . . 5  |-  ( Lim  x  ->  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  ( aleph `  x )  e.  On ) )
223, 5, 7, 5, 10, 14, 21tfinds 6679 . . . 4  |-  ( y  e.  On  ->  ( aleph `  y )  e.  On )
2322rgen 2803 . . 3  |-  A. y  e.  On  ( aleph `  y
)  e.  On
24 ffnfv 6042 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. y  e.  On  ( aleph `  y
)  e.  On ) )
251, 23, 24mpbir2an 920 . 2  |-  aleph : On --> On
26 0elon 4921 . 2  |-  (/)  e.  On
2725, 26f0cli 6027 1  |-  ( aleph `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095   (/)c0 3770   U_ciun 4315   Oncon0 4868   Lim wlim 4869   suc csuc 4870    Fn wfn 5573   -->wf 5574   ` cfv 5578   omcom 6685  harchar 7985   alephcale 8320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-om 6686  df-recs 7044  df-rdg 7078  df-en 7519  df-dom 7520  df-oi 7938  df-har 7987  df-aleph 8324
This theorem is referenced by:  alephnbtwn  8455  alephnbtwn2  8456  alephordilem1  8457  alephord  8459  alephord2  8460  alephord3  8462  alephsucdom  8463  alephsuc2  8464  alephf1  8469  alephsdom  8470  alephdom2  8471  alephle  8472  cardaleph  8473  alephf1ALT  8487  alephfp  8492  dfac12k  8530  alephsing  8659  alephval2  8950  alephadd  8955  alephmul  8956  alephexp1  8957  alephsuc3  8958  alephreg  8960  pwcfsdom  8961  cfpwsdom  8962  gchaleph  9052  gchaleph2  9053  gch2  9056
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