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

Theorem alephsuc 8929
Description: Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 8504, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephsuc (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))

Proof of Theorem alephsuc
StepHypRef Expression
1 rdgsuc 7565 . 2 (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴)))
2 df-aleph 8804 . . 3 ℵ = rec(har, ω)
32fveq1i 6230 . 2 (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴)
42fveq1i 6230 . . 3 (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
54fveq2i 6232 . 2 (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴))
61, 3, 53eqtr4g 2710 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Oncon0 5761  suc csuc 5763  cfv 5926  ωcom 7107  reccrdg 7550  harchar 8502  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-aleph 8804
This theorem is referenced by:  alephon  8930  alephcard  8931  alephnbtwn  8932  alephordilem1  8934  cardaleph  8950  gchaleph2  9532
  Copyright terms: Public domain W3C validator