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Theorem alephon 8446
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 8442 . . 3  |-  aleph  Fn  On
2 fveq2 5864 . . . . . 6  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32eleq1d 2536 . . . . 5  |-  ( x  =  (/)  ->  ( (
aleph `  x )  e.  On  <->  ( aleph `  (/) )  e.  On ) )
4 fveq2 5864 . . . . . 6  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54eleq1d 2536 . . . . 5  |-  ( x  =  y  ->  (
( aleph `  x )  e.  On  <->  ( aleph `  y
)  e.  On ) )
6 fveq2 5864 . . . . . 6  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
76eleq1d 2536 . . . . 5  |-  ( x  =  suc  y  -> 
( ( aleph `  x
)  e.  On  <->  ( aleph ` 
suc  y )  e.  On ) )
8 aleph0 8443 . . . . . 6  |-  ( aleph `  (/) )  =  om
9 omelon 8059 . . . . . 6  |-  om  e.  On
108, 9eqeltri 2551 . . . . 5  |-  ( aleph `  (/) )  e.  On
11 alephsuc 8445 . . . . . . 7  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
12 harcl 7983 . . . . . . 7  |-  (har `  ( aleph `  y )
)  e.  On
1311, 12syl6eqel 2563 . . . . . 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 3116 . . . . . . 7  |-  x  e. 
_V
16 fvex 5874 . . . . . . 7  |-  ( aleph `  y )  e.  _V
1715, 16iunonOLD 7007 . . . . . 6  |-  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  U_ y  e.  x  ( aleph `  y
)  e.  On )
18 alephlim 8444 . . . . . . . 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 2536 . . . . . 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 6672 . . . 4  |-  ( y  e.  On  ->  ( aleph `  y )  e.  On )
2322rgen 2824 . . 3  |-  A. y  e.  On  ( aleph `  y
)  e.  On
24 ffnfv 6045 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. y  e.  On  ( aleph `  y
)  e.  On ) )
251, 23, 24mpbir2an 918 . 2  |-  aleph : On --> On
26 0elon 4931 . 2  |-  (/)  e.  On
2725, 26f0cli 6030 1  |-  ( aleph `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   (/)c0 3785   U_ciun 4325   Oncon0 4878   Lim wlim 4879   suc csuc 4880    Fn wfn 5581   -->wf 5582   ` cfv 5586   omcom 6678  harchar 7978   alephcale 8313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-om 6679  df-recs 7039  df-rdg 7073  df-en 7514  df-dom 7515  df-oi 7931  df-har 7980  df-aleph 8317
This theorem is referenced by:  alephnbtwn  8448  alephnbtwn2  8449  alephordilem1  8450  alephord  8452  alephord2  8453  alephord3  8455  alephsucdom  8456  alephsuc2  8457  alephf1  8462  alephsdom  8463  alephdom2  8464  alephle  8465  cardaleph  8466  alephf1ALT  8480  alephfp  8485  dfac12k  8523  alephsing  8652  alephval2  8943  alephadd  8948  alephmul  8949  alephexp1  8950  alephsuc3  8951  alephreg  8953  pwcfsdom  8954  cfpwsdom  8955  gchaleph  9045  gchaleph2  9046  gch2  9049
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