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Theorem suceloni 4177
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 On → suc A On)

Proof of Theorem suceloni
StepHypRef Expression
1 eloni 4061 . . 3 (A On → Ord A)
2 ordsucim 4176 . . 3 (Ord A → Ord suc A)
31, 2syl 14 . 2 (A On → Ord suc A)
4 sucexg 4174 . . 3 (A On → suc A V)
5 elong 4059 . . 3 (suc A V → (suc A On ↔ Ord suc A))
64, 5syl 14 . 2 (A On → (suc A On ↔ Ord suc A))
73, 6mpbird 156 1 (A On → suc A On)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1374  Vcvv 2535  Ord word 4048  Oncon0 4049  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by:  sucelon  4179  unon  4186  onsuci  4191  ordsucunielexmid  4200  tfrlemisucaccv  5860  tfrexlem  5870  rdgisuc1  5891  frecsuclemdm  5904  oacl  5955  oasuc  5959  omsuc  5966
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