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Theorem suceloni 4485
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 4360 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordsucim 4484 . . 3  |-  ( Ord 
A  ->  Ord  suc  A
)
31, 2syl 14 . 2  |-  ( A  e.  On  ->  Ord  suc 
A )
4 sucexg 4482 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
5 elong 4358 . . 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 166 1  |-  ( A  e.  On  ->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 2141   _Vcvv 2730   Ord word 4347   Oncon0 4348   suc csuc 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by:  sucelon  4487  unon  4495  onsuci  4500  ordsucunielexmid  4515  tfrlemisucaccv  6304  tfrexlem  6313  tfri1dALT  6330  rdgisuc1  6363  rdgon  6365  oacl  6439  oasuc  6443  omsuc  6451
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