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Theorem ordsucim 4520
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 4399 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 suctr 4442 . . 3  |-  ( Tr  A  ->  Tr  suc  A
)
31, 2syl 14 . 2  |-  ( Ord 
A  ->  Tr  suc  A
)
4 df-suc 4392 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
54eleq2i 2256 . . . . 5  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
6 elun 3291 . . . . 5  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
7 velsn 3627 . . . . . 6  |-  ( x  e.  { A }  <->  x  =  A )
87orbi2i 763 . . . . 5  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  ( x  e.  A  \/  x  =  A ) )
95, 6, 83bitri 206 . . . 4  |-  ( x  e.  suc  A  <->  ( x  e.  A  \/  x  =  A ) )
10 dford3 4388 . . . . . . . 8  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
1110simprbi 275 . . . . . . 7  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
12 df-ral 2473 . . . . . . 7  |-  ( A. x  e.  A  Tr  x 
<-> 
A. x ( x  e.  A  ->  Tr  x ) )
1311, 12sylib 122 . . . . . 6  |-  ( Ord 
A  ->  A. x
( x  e.  A  ->  Tr  x ) )
141319.21bi 1569 . . . . 5  |-  ( Ord 
A  ->  ( x  e.  A  ->  Tr  x
) )
15 treq 4125 . . . . . 6  |-  ( x  =  A  ->  ( Tr  x  <->  Tr  A )
)
161, 15syl5ibrcom 157 . . . . 5  |-  ( Ord 
A  ->  ( x  =  A  ->  Tr  x
) )
1714, 16jaod 718 . . . 4  |-  ( Ord 
A  ->  ( (
x  e.  A  \/  x  =  A )  ->  Tr  x ) )
189, 17biimtrid 152 . . 3  |-  ( Ord 
A  ->  ( x  e.  suc  A  ->  Tr  x ) )
1918ralrimiv 2562 . 2  |-  ( Ord 
A  ->  A. x  e.  suc  A Tr  x
)
20 dford3 4388 . 2  |-  ( Ord 
suc  A  <->  ( Tr  suc  A  /\  A. x  e. 
suc  A Tr  x
) )
213, 19, 20sylanbrc 417 1  |-  ( Ord 
A  ->  Ord  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 709   A.wal 1362    = wceq 1364    e. wcel 2160   A.wral 2468    u. cun 3142   {csn 3610   Tr wtr 4119   Ord word 4383   suc csuc 4386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3616  df-uni 3828  df-tr 4120  df-iord 4387  df-suc 4392
This theorem is referenced by:  onsuc  4521  ordsucg  4522  onsucsssucr  4529  ordtriexmidlem  4539  2ordpr  4544  ordsuc  4583  nnsucsssuc  6521
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