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Theorem suctr 4667
Description: The sucessor of a transitive class is transitive. The proof of http://www.virtualdeduction.com/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctr 4667 using completeusersproof, which is verified by the Metamath program. The proof of http://www.virtualdeduction.com/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 11-Apr-2009.) See suctrALT 4666 for the original proof before this revision. (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
suctr  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctr
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4660 . . . . . . 7  |-  A  C_  suc  A
2 id 21 . . . . . . . 8  |-  ( Tr  A  ->  Tr  A
)
3 id 21 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 447 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 21 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  A )
6 trel 4311 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
763impib 1152 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
87idi 2 . . . . . . . 8  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 4, 5, 8syl3an 1227 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
101, 9sseldi 3348 . . . . . 6  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
11103expia 1156 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
124adantr 453 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  y )
13 id 21 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
1413adantl 454 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  y  =  A )
1512, 14eleqtrd 2514 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
161, 15sseldi 3348 . . . . . . 7  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1716ex 425 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
1817adantl 454 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  =  A  ->  z  e. 
suc  A ) )
193simprd 451 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
20 elsuci 4649 . . . . . . 7  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
2119, 20syl 16 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
2221adantl 454 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  \/  y  =  A ) )
2311, 18, 22mpjaod 372 . . . 4  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2423ex 425 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2524alrimivv 1643 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
26 dftr2 4306 . . 3  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2726biimpri 199 . 2  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
2825, 27syl 16 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   Tr wtr 4304   suc csuc 4585
This theorem is referenced by:  dfon2lem3  25414  dfon2lem7  25418  dford3lem2  27100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-sn 3822  df-uni 4018  df-tr 4305  df-suc 4589
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