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Theorem ordsuc 4379
Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
Assertion
Ref Expression
ordsuc  |-  ( Ord 
A  <->  Ord  suc  A )

Proof of Theorem ordsuc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordsucim 4317 . 2  |-  ( Ord 
A  ->  Ord  suc  A
)
2 en2lp 4370 . . . . . . . . . 10  |-  -.  (
x  e.  A  /\  A  e.  x )
3 eleq1 2150 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
43biimpac 292 . . . . . . . . . . . 12  |-  ( ( y  e.  x  /\  y  =  A )  ->  A  e.  x )
54anim2i 334 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  ( y  e.  x  /\  y  =  A
) )  ->  (
x  e.  A  /\  A  e.  x )
)
65expr 367 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  x )  ->  ( y  =  A  ->  ( x  e.  A  /\  A  e.  x ) ) )
72, 6mtoi 625 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  x )  ->  -.  y  =  A )
87adantl 271 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  -.  y  =  A )
9 elelsuc 4236 . . . . . . . . . . . . . . 15  |-  ( x  e.  A  ->  x  e.  suc  A )
109adantr 270 . . . . . . . . . . . . . 14  |-  ( ( x  e.  A  /\  y  e.  x )  ->  x  e.  suc  A
)
11 ordelss 4206 . . . . . . . . . . . . . 14  |-  ( ( Ord  suc  A  /\  x  e.  suc  A )  ->  x  C_  suc  A )
1210, 11sylan2 280 . . . . . . . . . . . . 13  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  x  C_ 
suc  A )
1312sseld 3024 . . . . . . . . . . . 12  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  (
y  e.  x  -> 
y  e.  suc  A
) )
1413expr 367 . . . . . . . . . . 11  |-  ( ( Ord  suc  A  /\  x  e.  A )  ->  ( y  e.  x  ->  ( y  e.  x  ->  y  e.  suc  A
) ) )
1514pm2.43d 49 . . . . . . . . . 10  |-  ( ( Ord  suc  A  /\  x  e.  A )  ->  ( y  e.  x  ->  y  e.  suc  A
) )
1615impr 371 . . . . . . . . 9  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  y  e.  suc  A )
17 elsuci 4230 . . . . . . . . 9  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1816, 17syl 14 . . . . . . . 8  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  (
y  e.  A  \/  y  =  A )
)
198, 18ecased 1285 . . . . . . 7  |-  ( ( Ord  suc  A  /\  ( x  e.  A  /\  y  e.  x
) )  ->  y  e.  A )
2019ancom2s 533 . . . . . 6  |-  ( ( Ord  suc  A  /\  ( y  e.  x  /\  x  e.  A
) )  ->  y  e.  A )
2120ex 113 . . . . 5  |-  ( Ord 
suc  A  ->  ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
2221alrimivv 1803 . . . 4  |-  ( Ord 
suc  A  ->  A. y A. x ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
23 dftr2 3938 . . . 4  |-  ( Tr  A  <->  A. y A. x
( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
2422, 23sylibr 132 . . 3  |-  ( Ord 
suc  A  ->  Tr  A
)
25 sssucid 4242 . . . 4  |-  A  C_  suc  A
26 trssord 4207 . . . 4  |-  ( ( Tr  A  /\  A  C_ 
suc  A  /\  Ord  suc  A )  ->  Ord  A )
2725, 26mp3an2 1261 . . 3  |-  ( ( Tr  A  /\  Ord  suc 
A )  ->  Ord  A )
2824, 27mpancom 413 . 2  |-  ( Ord 
suc  A  ->  Ord  A
)
291, 28impbii 124 1  |-  ( Ord 
A  <->  Ord  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438    C_ wss 2999   Tr wtr 3936   Ord word 4189   suc csuc 4192
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-in1 579  ax-in2 580  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  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  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-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-suc 4198
This theorem is referenced by:  nlimsucg  4382  ordpwsucss  4383
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