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Theorem ordsucim 4307
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 4196 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 suctr 4239 . . 3  |-  ( Tr  A  ->  Tr  suc  A
)
31, 2syl 14 . 2  |-  ( Ord 
A  ->  Tr  suc  A
)
4 df-suc 4189 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54eleq2i 2154 . . . . 5  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
6 elun 3139 . . . . 5  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
7 velsn 3458 . . . . . 6  |-  ( x  e.  { A }  <->  x  =  A )
87orbi2i 714 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  ( x  e.  A  \/  x  =  A ) )
95, 6, 83bitri 204 . . . 4  |-  ( x  e.  suc  A  <->  ( x  e.  A  \/  x  =  A ) )
10 dford3 4185 . . . . . . . 8  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
1110simprbi 269 . . . . . . 7  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
12 df-ral 2364 . . . . . . 7  |-  ( A. x  e.  A  Tr  x 
<-> 
A. x ( x  e.  A  ->  Tr  x ) )
1311, 12sylib 120 . . . . . 6  |-  ( Ord 
A  ->  A. x
( x  e.  A  ->  Tr  x ) )
141319.21bi 1495 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  A  ->  Tr  x
) )
15 treq 3934 . . . . . 6  |-  ( x  =  A  ->  ( Tr  x  <->  Tr  A )
)
161, 15syl5ibrcom 155 . . . . 5  |-  ( Ord 
A  ->  ( x  =  A  ->  Tr  x
) )
1714, 16jaod 672 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  A  \/  x  =  A )  ->  Tr  x ) )
189, 17syl5bi 150 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  Tr  x ) )
1918ralrimiv 2445 . 2  |-  ( Ord 
A  ->  A. x  e.  suc  A Tr  x
)
20 dford3 4185 . 2  |-  ( Ord 
suc  A  <->  ( Tr  suc  A  /\  A. x  e. 
suc  A Tr  x
) )
213, 19, 20sylanbrc 408 1  |-  ( Ord 
A  ->  Ord  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359    u. cun 2995   {csn 3441   Tr wtr 3928   Ord word 4180   suc csuc 4183
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  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 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  suceloni  4308  ordsucg  4309  onsucsssucr  4316  ordtriexmidlem  4326  2ordpr  4330  ordsuc  4369  nnsucsssuc  6235
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