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Theorem ordsucim 4330
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
Assertion
Ref Expression
ordsucim  |-  ( Ord 
A  ->  Ord  suc  A
)

Proof of Theorem ordsucim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordtr 4214 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 suctr 4257 . . 3  |-  ( Tr  A  ->  Tr  suc  A
)
31, 2syl 14 . 2  |-  ( Ord 
A  ->  Tr  suc  A
)
4 df-suc 4207 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54eleq2i 2155 . . . . 5  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
6 elun 3142 . . . . 5  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
7 velsn 3467 . . . . . 6  |-  ( x  e.  { A }  <->  x  =  A )
87orbi2i 715 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  ( x  e.  A  \/  x  =  A ) )
95, 6, 83bitri 205 . . . 4  |-  ( x  e.  suc  A  <->  ( x  e.  A  \/  x  =  A ) )
10 dford3 4203 . . . . . . . 8  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
1110simprbi 270 . . . . . . 7  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
12 df-ral 2365 . . . . . . 7  |-  ( A. x  e.  A  Tr  x 
<-> 
A. x ( x  e.  A  ->  Tr  x ) )
1311, 12sylib 121 . . . . . 6  |-  ( Ord 
A  ->  A. x
( x  e.  A  ->  Tr  x ) )
141319.21bi 1496 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  A  ->  Tr  x
) )
15 treq 3948 . . . . . 6  |-  ( x  =  A  ->  ( Tr  x  <->  Tr  A )
)
161, 15syl5ibrcom 156 . . . . 5  |-  ( Ord 
A  ->  ( x  =  A  ->  Tr  x
) )
1714, 16jaod 673 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  A  \/  x  =  A )  ->  Tr  x ) )
189, 17syl5bi 151 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  Tr  x ) )
1918ralrimiv 2446 . 2  |-  ( Ord 
A  ->  A. x  e.  suc  A Tr  x
)
20 dford3 4203 . 2  |-  ( Ord 
suc  A  <->  ( Tr  suc  A  /\  A. x  e. 
suc  A Tr  x
) )
213, 19, 20sylanbrc 409 1  |-  ( Ord 
A  ->  Ord  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 665   A.wal 1288    = wceq 1290    e. wcel 1439   A.wral 2360    u. cun 2998   {csn 3450   Tr wtr 3942   Ord word 4198   suc csuc 4201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202  df-suc 4207
This theorem is referenced by:  suceloni  4331  ordsucg  4332  onsucsssucr  4339  ordtriexmidlem  4349  2ordpr  4353  ordsuc  4392  nnsucsssuc  6267
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