Theorem alephsuc 10065

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 9554, 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 8425	. 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴)))
2		df-aleph 9937	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6886	. 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴)
4	2	fveq1i 6886	. . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
5	4	fveq2i 6888	. 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴))
6	1, 3, 5	3eqtr4g 2791	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Oncon0 6358  suc csuc 6360  ‘cfv 6537  ωcom 7852  reccrdg 8410  harchar 9553  ℵcale 9933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-aleph 9937

This theorem is referenced by:  alephon  10066  alephcard  10067  alephnbtwn  10068  alephordilem1  10070  cardaleph  10086  gchaleph2  10669  aleph1min  42881