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Theorem suceloni 4318
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
suceloni  |-  ( A  e.  On  ->  suc  A  e.  On )

Proof of Theorem suceloni
StepHypRef Expression
1 eloni 4202 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordsucim 4317 . . 3  |-  ( Ord 
A  ->  Ord  suc  A
)
31, 2syl 14 . 2  |-  ( A  e.  On  ->  Ord  suc 
A )
4 sucexg 4315 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
5 elong 4200 . . 3  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  On  <->  Ord 
suc  A ) )
64, 5syl 14 . 2  |-  ( A  e.  On  ->  ( suc  A  e.  On  <->  Ord  suc  A
) )
73, 6mpbird 165 1  |-  ( A  e.  On  ->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   _Vcvv 2619   Ord word 4189   Oncon0 4190   suc csuc 4192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198
This theorem is referenced by:  sucelon  4320  unon  4328  onsuci  4333  ordsucunielexmid  4347  tfrlemisucaccv  6090  tfrexlem  6099  tfri1dALT  6116  rdgisuc1  6149  rdgon  6151  oacl  6221  oasuc  6225  omsuc  6233
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