Theorem alephsuc 9493

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 9021, 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

Step	Hyp	Ref	Expression
1		rdgsuc 8059	. 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴)))
2		df-aleph 9368	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6670	. 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴)
4	2	fveq1i 6670	. . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
5	4	fveq2i 6672	. 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴))
6	1, 3, 5	3eqtr4g 2881	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Oncon0 6190  suc csuc 6192  ‘cfv 6354  ωcom 7579  reccrdg 8044  harchar 9019  ℵcale 9364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-aleph 9368

This theorem is referenced by:  alephon  9494  alephcard  9495  alephnbtwn  9496  alephordilem1  9498  cardaleph  9514  gchaleph2  10093  aleph1min  39914