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Theorem ordsucim 4386
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 4270 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 suctr 4313 . . 3  |-  ( Tr  A  ->  Tr  suc  A
)
31, 2syl 14 . 2  |-  ( Ord 
A  ->  Tr  suc  A
)
4 df-suc 4263 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54eleq2i 2184 . . . . 5  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
6 elun 3187 . . . . 5  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
7 velsn 3514 . . . . . 6  |-  ( x  e.  { A }  <->  x  =  A )
87orbi2i 736 . . . . 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 4259 . . . . . . . 8  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
1110simprbi 273 . . . . . . 7  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
12 df-ral 2398 . . . . . . 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 1522 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  A  ->  Tr  x
) )
15 treq 4002 . . . . . 6  |-  ( x  =  A  ->  ( Tr  x  <->  Tr  A )
)
161, 15syl5ibrcom 156 . . . . 5  |-  ( Ord 
A  ->  ( x  =  A  ->  Tr  x
) )
1714, 16jaod 691 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  A  \/  x  =  A )  ->  Tr  x ) )
189, 17syl5bi 151 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  Tr  x ) )
1918ralrimiv 2481 . 2  |-  ( Ord 
A  ->  A. x  e.  suc  A Tr  x
)
20 dford3 4259 . 2  |-  ( Ord 
suc  A  <->  ( Tr  suc  A  /\  A. x  e. 
suc  A Tr  x
) )
213, 19, 20sylanbrc 413 1  |-  ( Ord 
A  ->  Ord  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 682   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393    u. cun 3039   {csn 3497   Tr wtr 3996   Ord word 4254   suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-suc 4263
This theorem is referenced by:  suceloni  4387  ordsucg  4388  onsucsssucr  4395  ordtriexmidlem  4405  2ordpr  4409  ordsuc  4448  nnsucsssuc  6356
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