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Theorem suceloni 4193
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 4078 . . 3 (A On → Ord A)
2 ordsucim 4192 . . 3 (Ord A → Ord suc A)
31, 2syl 14 . 2 (A On → Ord suc A)
4 sucexg 4190 . . 3 (A On → suc A V)
5 elong 4076 . . 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 1390  Vcvv 2551  Ord word 4065  Oncon0 4066  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by:  sucelon  4195  unon  4202  onsuci  4207  ordsucunielexmid  4216  tfrlemisucaccv  5880  tfrexlem  5889  rdgisuc1  5911  frecsuclemdm  5927  oacl  5979  oasuc  5983  omsuc  5990
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