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Theorem sucelon 4286
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 4284 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
2 eloni 4169 . . 3  |-  ( suc 
A  e.  On  ->  Ord 
suc  A )
3 elex 2623 . . . . 5  |-  ( suc 
A  e.  On  ->  suc 
A  e.  _V )
4 sucexb 4280 . . . . 5  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
53, 4sylibr 132 . . . 4  |-  ( suc 
A  e.  On  ->  A  e.  _V )
6 elong 4167 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
7 ordsucg 4285 . . . . 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 1436   _Vcvv 2614   Ord word 4156   Oncon0 4157   suc csuc 4159
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165
This theorem is referenced by:  onsucmin  4290  onsucuni2  4346
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