Theorem alephsuc 9962

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 9449, 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 8346	. 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴)))
2		df-aleph 9836	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6823	. 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴)
4	2	fveq1i 6823	. . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
5	4	fveq2i 6825	. 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴))
6	1, 3, 5	3eqtr4g 2789	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Oncon0 6307  suc csuc 6309  ‘cfv 6482  ωcom 7799  reccrdg 8331  harchar 9448  ℵcale 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-aleph 9836

This theorem is referenced by:  alephon  9963  alephcard  9964  alephnbtwn  9965  alephordilem1  9967  cardaleph  9983  gchaleph2  10566  aleph1min  43534