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Theorem alephsuc 10099
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 9588, 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 β†’ (β„΅β€˜suc 𝐴) = (harβ€˜(β„΅β€˜π΄)))

Proof of Theorem alephsuc
StepHypRef Expression
1 rdgsuc 8451 . 2 (𝐴 ∈ On β†’ (rec(har, Ο‰)β€˜suc 𝐴) = (harβ€˜(rec(har, Ο‰)β€˜π΄)))
2 df-aleph 9971 . . 3 β„΅ = rec(har, Ο‰)
32fveq1i 6903 . 2 (β„΅β€˜suc 𝐴) = (rec(har, Ο‰)β€˜suc 𝐴)
42fveq1i 6903 . . 3 (β„΅β€˜π΄) = (rec(har, Ο‰)β€˜π΄)
54fveq2i 6905 . 2 (harβ€˜(β„΅β€˜π΄)) = (harβ€˜(rec(har, Ο‰)β€˜π΄))
61, 3, 53eqtr4g 2793 1 (𝐴 ∈ On β†’ (β„΅β€˜suc 𝐴) = (harβ€˜(β„΅β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Oncon0 6374  suc csuc 6376  β€˜cfv 6553  Ο‰com 7876  reccrdg 8436  harchar 9587  β„΅cale 9967
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-aleph 9971
This theorem is referenced by:  alephon  10100  alephcard  10101  alephnbtwn  10102  alephordilem1  10104  cardaleph  10120  gchaleph2  10703  aleph1min  43018
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