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Theorem ordsucim 4254
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 4143 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 suctr 4186 . . 3  |-  ( Tr  A  ->  Tr  suc  A
)
31, 2syl 14 . 2  |-  ( Ord 
A  ->  Tr  suc  A
)
4 df-suc 4136 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54eleq2i 2120 . . . . 5  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
6 elun 3112 . . . . 5  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
7 velsn 3420 . . . . . 6  |-  ( x  e.  { A }  <->  x  =  A )
87orbi2i 689 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  ( x  e.  A  \/  x  =  A ) )
95, 6, 83bitri 199 . . . 4  |-  ( x  e.  suc  A  <->  ( x  e.  A  \/  x  =  A ) )
10 dford3 4132 . . . . . . . 8  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
1110simprbi 264 . . . . . . 7  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
12 df-ral 2328 . . . . . . 7  |-  ( A. x  e.  A  Tr  x 
<-> 
A. x ( x  e.  A  ->  Tr  x ) )
1311, 12sylib 131 . . . . . 6  |-  ( Ord 
A  ->  A. x
( x  e.  A  ->  Tr  x ) )
141319.21bi 1466 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  A  ->  Tr  x
) )
15 treq 3888 . . . . . 6  |-  ( x  =  A  ->  ( Tr  x  <->  Tr  A )
)
161, 15syl5ibrcom 150 . . . . 5  |-  ( Ord 
A  ->  ( x  =  A  ->  Tr  x
) )
1714, 16jaod 647 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  A  \/  x  =  A )  ->  Tr  x ) )
189, 17syl5bi 145 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  Tr  x ) )
1918ralrimiv 2408 . 2  |-  ( Ord 
A  ->  A. x  e.  suc  A Tr  x
)
20 dford3 4132 . 2  |-  ( Ord 
suc  A  <->  ( Tr  suc  A  /\  A. x  e. 
suc  A Tr  x
) )
213, 19, 20sylanbrc 402 1  |-  ( Ord 
A  ->  Ord  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323    u. cun 2943   {csn 3403   Tr wtr 3882   Ord word 4127   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-uni 3609  df-tr 3883  df-iord 4131  df-suc 4136
This theorem is referenced by:  suceloni  4255  ordsucg  4256  onsucsssucr  4263  ordtriexmidlem  4273  2ordpr  4277  ordsuc  4315  nnsucsssuc  6102
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