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Theorem alephon 8237
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 8233 . . 3  |-  aleph  Fn  On
2 fveq2 5689 . . . . . 6  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32eleq1d 2507 . . . . 5  |-  ( x  =  (/)  ->  ( (
aleph `  x )  e.  On  <->  ( aleph `  (/) )  e.  On ) )
4 fveq2 5689 . . . . . 6  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54eleq1d 2507 . . . . 5  |-  ( x  =  y  ->  (
( aleph `  x )  e.  On  <->  ( aleph `  y
)  e.  On ) )
6 fveq2 5689 . . . . . 6  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
76eleq1d 2507 . . . . 5  |-  ( x  =  suc  y  -> 
( ( aleph `  x
)  e.  On  <->  ( aleph ` 
suc  y )  e.  On ) )
8 aleph0 8234 . . . . . 6  |-  ( aleph `  (/) )  =  om
9 omelon 7850 . . . . . 6  |-  om  e.  On
108, 9eqeltri 2511 . . . . 5  |-  ( aleph `  (/) )  e.  On
11 alephsuc 8236 . . . . . . 7  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
12 harcl 7774 . . . . . . 7  |-  (har `  ( aleph `  y )
)  e.  On
1311, 12syl6eqel 2529 . . . . . 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 2973 . . . . . . 7  |-  x  e. 
_V
16 fvex 5699 . . . . . . 7  |-  ( aleph `  y )  e.  _V
1715, 16iunonOLD 6798 . . . . . 6  |-  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  U_ y  e.  x  ( aleph `  y
)  e.  On )
18 alephlim 8235 . . . . . . . 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 2507 . . . . . 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 6468 . . . 4  |-  ( y  e.  On  ->  ( aleph `  y )  e.  On )
2322rgen 2779 . . 3  |-  A. y  e.  On  ( aleph `  y
)  e.  On
24 ffnfv 5867 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. y  e.  On  ( aleph `  y
)  e.  On ) )
251, 23, 24mpbir2an 911 . 2  |-  aleph : On --> On
26 0elon 4770 . 2  |-  (/)  e.  On
2725, 26f0cli 5852 1  |-  ( aleph `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   A.wral 2713   _Vcvv 2970   (/)c0 3635   U_ciun 4169   Oncon0 4717   Lim wlim 4718   suc csuc 4719    Fn wfn 5411   -->wf 5412   ` cfv 5416   omcom 6474  harchar 7769   alephcale 8104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-om 6475  df-recs 6830  df-rdg 6864  df-en 7309  df-dom 7310  df-oi 7722  df-har 7771  df-aleph 8108
This theorem is referenced by:  alephnbtwn  8239  alephnbtwn2  8240  alephordilem1  8241  alephord  8243  alephord2  8244  alephord3  8246  alephsucdom  8247  alephsuc2  8248  alephf1  8253  alephsdom  8254  alephdom2  8255  alephle  8256  cardaleph  8257  alephf1ALT  8271  alephfp  8276  dfac12k  8314  alephsing  8443  alephval2  8734  alephadd  8739  alephmul  8740  alephexp1  8741  alephsuc3  8742  alephreg  8744  pwcfsdom  8745  cfpwsdom  8746  gchaleph  8836  gchaleph2  8837  gch2  8840
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