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Theorem alephsuc 7664
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 7241, 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 6406 . 2  |-  ( A  e.  On  ->  ( rec (har ,  om ) `  suc  A )  =  (har `  ( rec (har ,  om ) `  A ) ) )
2 df-aleph 7542 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5460 . 2  |-  ( aleph ` 
suc  A )  =  ( rec (har ,  om ) `  suc  A
)
42fveq1i 5460 . . 3  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
54fveq2i 5462 . 2  |-  (har `  ( aleph `  A )
)  =  (har `  ( rec (har ,  om ) `  A )
)
61, 3, 53eqtr4g 2315 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   Oncon0 4365   suc csuc 4367   omcom 4629   ` cfv 5195   reccrdg 6391  harchar 7239   alephcale 7538
This theorem is referenced by:  alephon  7665  alephcard  7666  alephnbtwn  7667  alephordilem1  7669  cardaleph  7685  gchaleph2  8267
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4106  ax-sep 4116  ax-nul 4124  ax-pr 4187  ax-un 4485
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3541  df-pw 3602  df-sn 3621  df-pr 3622  df-tp 3623  df-op 3624  df-uni 3803  df-iun 3882  df-br 3999  df-opab 4053  df-mpt 4054  df-tr 4089  df-eprel 4278  df-id 4282  df-po 4287  df-so 4288  df-fr 4325  df-we 4327  df-ord 4368  df-on 4369  df-lim 4370  df-suc 4371  df-xp 4668  df-rel 4669  df-cnv 4670  df-co 4671  df-dm 4672  df-rn 4673  df-res 4674  df-ima 4675  df-fun 5197  df-fn 5198  df-f 5199  df-f1 5200  df-fo 5201  df-f1o 5202  df-fv 5203  df-recs 6357  df-rdg 6392  df-aleph 7542
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