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Theorem sucelon 4318
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
Assertion
Ref Expression
sucelon  |-  ( A  e.  On  <->  suc  A  e.  On )

Proof of Theorem sucelon
StepHypRef Expression
1 suceloni 4316 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 eloni 4200 . . 3  |-  ( suc 
A  e.  On  ->  Ord 
suc  A )
3 elex 2630 . . . . 5  |-  ( suc 
A  e.  On  ->  suc 
A  e.  _V )
4 sucexb 4312 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
53, 4sylibr 132 . . . 4  |-  ( suc 
A  e.  On  ->  A  e.  _V )
6 elong 4198 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
7 ordsucg 4317 . . . . 5  |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
)
86, 7bitrd 186 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  suc  A
) )
95, 8syl 14 . . 3  |-  ( suc 
A  e.  On  ->  ( A  e.  On  <->  Ord  suc  A
) )
102, 9mpbird 165 . 2  |-  ( suc 
A  e.  On  ->  A  e.  On )
111, 10impbii 124 1  |-  ( A  e.  On  <->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   _Vcvv 2619   Ord word 4187   Oncon0 4188   suc csuc 4190
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 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  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 3429  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-iord 4191  df-on 4193  df-suc 4196
This theorem is referenced by:  onsucmin  4322  onsucuni2  4378
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