MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephfnon Structured version   Visualization version   GIF version

Theorem alephfnon 8832
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 7459 . 2 rec(har, ω) Fn On
2 df-aleph 8710 . . 3 ℵ = rec(har, ω)
32fneq1i 5943 . 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 5682   Fn wfn 5842  ωcom 7012  reccrdg 7450  harchar 8405  cale 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-aleph 8710
This theorem is referenced by:  alephon  8836  alephcard  8837  alephnbtwn  8838  alephgeom  8849  alephf1  8852  infenaleph  8858  isinfcard  8859  alephiso  8865  alephsmo  8869  alephf1ALT  8870  alephfplem1  8871  alephfplem3  8873  alephsing  9042  alephadd  9343  alephreg  9348  pwcfsdom  9349  cfpwsdom  9350  gch2  9441  gch3  9442
  Copyright terms: Public domain W3C validator