Theorem alephsuc 9824

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 9316, 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 8255	. 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴)))
2		df-aleph 9698	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6775	. 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴)
4	2	fveq1i 6775	. . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
5	4	fveq2i 6777	. 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴))
6	1, 3, 5	3eqtr4g 2803	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Oncon0 6266  suc csuc 6268  ‘cfv 6433  ωcom 7712  reccrdg 8240  harchar 9315  ℵcale 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-aleph 9698

This theorem is referenced by:  alephon  9825  alephcard  9826  alephnbtwn  9827  alephordilem1  9829  cardaleph  9845  gchaleph2  10428  aleph1min  41164