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Theorem alephfnon 8873
Description: The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
alephfnon ℵ Fn On

Proof of Theorem alephfnon
StepHypRef Expression
1 rdgfnon 7499 . 2 rec(har, ω) Fn On
2 df-aleph 8751 . . 3 ℵ = rec(har, ω)
32fneq1i 5973 . 2 (ℵ Fn On ↔ rec(har, ω) Fn On)
41, 3mpbir 221 1 ℵ Fn On
Colors of variables: wff setvar class
Syntax hints:  Oncon0 5711   Fn wfn 5871  ωcom 7050  reccrdg 7490  harchar 8446  cale 8747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-aleph 8751
This theorem is referenced by:  alephon  8877  alephcard  8878  alephnbtwn  8879  alephgeom  8890  alephf1  8893  infenaleph  8899  isinfcard  8900  alephiso  8906  alephsmo  8910  alephf1ALT  8911  alephfplem1  8912  alephfplem3  8914  alephsing  9083  alephadd  9384  alephreg  9389  pwcfsdom  9390  cfpwsdom  9391  gch2  9482  gch3  9483
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