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Theorem alephsuc 8470
 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 8005, 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

Proof of Theorem alephsuc
StepHypRef Expression
1 rdgsuc 7109 . 2
2 df-aleph 8342 . . 3
32fveq1i 5872 . 2
42fveq1i 5872 . . 3
54fveq2i 5874 . 2
61, 3, 53eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818   con0 4883  succsuc 4885  cfv 5593   com 6700  rec`crdg 7094   char 8003   cale 8338 This theorem is referenced by:  alephon  8471  alephcard  8472  alephnbtwn  8473  alephordilem1  8475  cardaleph  8491  gchaleph2  9071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-recs 7061  df-rdg 7095  df-aleph 8342
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