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Theorem alephsuc 7842
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 7419, 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 6579 . 2  |-  ( A  e.  On  ->  ( rec (har ,  om ) `  suc  A )  =  (har `  ( rec (har ,  om ) `  A ) ) )
2 df-aleph 7720 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5633 . 2  |-  ( aleph ` 
suc  A )  =  ( rec (har ,  om ) `  suc  A
)
42fveq1i 5633 . . 3  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
54fveq2i 5635 . 2  |-  (har `  ( aleph `  A )
)  =  (har `  ( rec (har ,  om ) `  A )
)
61, 3, 53eqtr4g 2423 1  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   Oncon0 4495   suc csuc 4497   omcom 4759   ` cfv 5358   reccrdg 6564  harchar 7417   alephcale 7716
This theorem is referenced by:  alephon  7843  alephcard  7844  alephnbtwn  7845  alephordilem1  7847  cardaleph  7863  gchaleph2  8445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-recs 6530  df-rdg 6565  df-aleph 7720
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