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Theorem alephsuc 7697
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 7274, 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 6439 . 2  |-  ( A  e.  On  ->  ( rec (har ,  om ) `  suc  A )  =  (har `  ( rec (har ,  om ) `  A ) ) )
2 df-aleph 7575 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5528 . 2  |-  ( aleph ` 
suc  A )  =  ( rec (har ,  om ) `  suc  A
)
42fveq1i 5528 . . 3  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
54fveq2i 5530 . 2  |-  (har `  ( aleph `  A )
)  =  (har `  ( rec (har ,  om ) `  A )
)
61, 3, 53eqtr4g 2342 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   Oncon0 4394   suc csuc 4396   omcom 4658   ` cfv 5257   reccrdg 6424  harchar 7272   alephcale 7571
This theorem is referenced by:  alephon  7698  alephcard  7699  alephnbtwn  7700  alephordilem1  7702  cardaleph  7718  gchaleph2  8300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-aleph 7575
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