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Theorem suctrALT 33107
Description: The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 33107 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 4967 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALT
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4961 . . . . . . 7  |-  A  C_  suc  A
2 id 22 . . . . . . . 8  |-  ( Tr  A  ->  Tr  A
)
3 id 22 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 459 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 22 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  A )
6 trel 4553 . . . . . . . . . 10  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
763impib 1194 . . . . . . . . 9  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
87idiALT 32698 . . . . . . . 8  |-  ( ( Tr  A  /\  z  e.  y  /\  y  e.  A )  ->  z  e.  A )
92, 4, 5, 8syl3an 1270 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
101, 9sseldi 3507 . . . . . 6  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
11103expia 1198 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
124adantr 465 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  y )
13 id 22 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
1413adantl 466 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  y  =  A )
1512, 14eleqtrd 2557 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
161, 15sseldi 3507 . . . . . . 7  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1716ex 434 . . . . . 6  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
1817adantl 466 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  =  A  ->  z  e. 
suc  A ) )
193simprd 463 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
20 elsuci 4950 . . . . . . 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 466 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  \/  y  =  A ) )
2311, 18, 22mpjaod 381 . . . 4  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2423ex 434 . . 3  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2524alrimivv 1696 . 2  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
26 dftr2 4548 . . 3  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2726biimpri 206 . 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   Tr wtr 4546   suc csuc 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495  df-sn 4034  df-uni 4252  df-tr 4547  df-suc 4890
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator