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Theorem alephon 7950
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 7946 . . 3  |-  aleph  Fn  On
2 fveq2 5728 . . . . . 6  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32eleq1d 2502 . . . . 5  |-  ( x  =  (/)  ->  ( (
aleph `  x )  e.  On  <->  ( aleph `  (/) )  e.  On ) )
4 fveq2 5728 . . . . . 6  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
54eleq1d 2502 . . . . 5  |-  ( x  =  y  ->  (
( aleph `  x )  e.  On  <->  ( aleph `  y
)  e.  On ) )
6 fveq2 5728 . . . . . 6  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
76eleq1d 2502 . . . . 5  |-  ( x  =  suc  y  -> 
( ( aleph `  x
)  e.  On  <->  ( aleph ` 
suc  y )  e.  On ) )
8 aleph0 7947 . . . . . 6  |-  ( aleph `  (/) )  =  om
9 omelon 7601 . . . . . 6  |-  om  e.  On
108, 9eqeltri 2506 . . . . 5  |-  ( aleph `  (/) )  e.  On
11 alephsuc 7949 . . . . . . 7  |-  ( y  e.  On  ->  ( aleph `  suc  y )  =  (har `  ( aleph `  y ) ) )
12 harcl 7529 . . . . . . 7  |-  (har `  ( aleph `  y )
)  e.  On
1311, 12syl6eqel 2524 . . . . . 6  |-  ( y  e.  On  ->  ( aleph `  suc  y )  e.  On )
1413a1d 23 . . . . 5  |-  ( y  e.  On  ->  (
( aleph `  y )  e.  On  ->  ( aleph ` 
suc  y )  e.  On ) )
15 vex 2959 . . . . . . 7  |-  x  e. 
_V
16 fvex 5742 . . . . . . 7  |-  ( aleph `  y )  e.  _V
1715, 16iunonOLD 6601 . . . . . 6  |-  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  U_ y  e.  x  ( aleph `  y
)  e.  On )
18 alephlim 7948 . . . . . . . 8  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ y  e.  x  ( aleph `  y )
)
1915, 18mpan 652 . . . . . . 7  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ y  e.  x  ( aleph `  y ) )
2019eleq1d 2502 . . . . . 6  |-  ( Lim  x  ->  ( ( aleph `  x )  e.  On  <->  U_ y  e.  x  ( aleph `  y )  e.  On ) )
2117, 20syl5ibr 213 . . . . 5  |-  ( Lim  x  ->  ( A. y  e.  x  ( aleph `  y )  e.  On  ->  ( aleph `  x )  e.  On ) )
223, 5, 7, 5, 10, 14, 21tfinds 4839 . . . 4  |-  ( y  e.  On  ->  ( aleph `  y )  e.  On )
2322rgen 2771 . . 3  |-  A. y  e.  On  ( aleph `  y
)  e.  On
24 ffnfv 5894 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. y  e.  On  ( aleph `  y
)  e.  On ) )
251, 23, 24mpbir2an 887 . 2  |-  aleph : On --> On
26 0elon 4634 . 2  |-  (/)  e.  On
2725, 26f0cli 5880 1  |-  ( aleph `  A )  e.  On
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   (/)c0 3628   U_ciun 4093   Oncon0 4581   Lim wlim 4582   suc csuc 4583   omcom 4845    Fn wfn 5449   -->wf 5450   ` cfv 5454  harchar 7524   alephcale 7823
This theorem is referenced by:  alephnbtwn  7952  alephnbtwn2  7953  alephordilem1  7954  alephord  7956  alephord2  7957  alephord3  7959  alephsucdom  7960  alephsuc2  7961  alephf1  7966  alephsdom  7967  alephdom2  7968  alephle  7969  cardaleph  7970  alephf1ALT  7984  alephfp  7989  dfac12k  8027  alephsing  8156  alephval2  8447  alephadd  8452  alephmul  8453  alephexp1  8454  alephsuc3  8455  alephreg  8457  pwcfsdom  8458  cfpwsdom  8459  gchaleph  8550  gchaleph2  8551  gch2  8554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-rdg 6668  df-en 7110  df-dom 7111  df-oi 7479  df-har 7526  df-aleph 7827
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