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Theorem alephsuc 8399
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 7936, 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  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )

Proof of Theorem alephsuc
StepHypRef Expression
1 rdgsuc 7045 . 2  |-  ( A  e.  On  ->  ( rec (har ,  om ) `  suc  A )  =  (har `  ( rec (har ,  om ) `  A ) ) )
2 df-aleph 8271 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5804 . 2  |-  ( aleph ` 
suc  A )  =  ( rec (har ,  om ) `  suc  A
)
42fveq1i 5804 . . 3  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
54fveq2i 5806 . 2  |-  (har `  ( aleph `  A )
)  =  (har `  ( rec (har ,  om ) `  A )
)
61, 3, 53eqtr4g 2466 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   Oncon0 4819   suc csuc 4821   ` cfv 5523   omcom 6636   reccrdg 7030  harchar 7934   alephcale 8267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-recs 6997  df-rdg 7031  df-aleph 8271
This theorem is referenced by:  alephon  8400  alephcard  8401  alephnbtwn  8402  alephordilem1  8404  cardaleph  8420  gchaleph2  8998
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