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Theorem suctrALT3 36735
Description: The successor of a transtive class is transitive. suctrALT3 36735 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 36350 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction ( e.g. , the sub-theorem whose assertion is step 19 used jaoded 36343). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem ( e.g. , the sub-theorem whose assertion is step 24 used dftr2 4490) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT3  |-  ( Tr  A  ->  Tr  suc  A
)

Proof of Theorem suctrALT3
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 5486 . . . . . . . . 9  |-  A  C_  suc  A
2 id 22 . . . . . . . . . 10  |-  ( Tr  A  ->  Tr  A
)
3 id 22 . . . . . . . . . . 11  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 457 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 22 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  A )
62, 4, 5trelded 36342 . . . . . . . . 9  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
71, 6sseldi 3439 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
873expia 1199 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
9 id 22 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
10 eleq2 2475 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1110biimpac 484 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
124, 9, 11syl2an 475 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
131, 12sseldi 3439 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1413ex 432 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
153simprd 461 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
16 elsuci 5475 . . . . . . . 8  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1715, 16syl 17 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
188, 14, 17jaoded 36343 . . . . . 6  |-  ( ( ( Tr  A  /\  ( z  e.  y  /\  y  e.  suc  A ) )  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  ( z  e.  y  /\  y  e. 
suc  A ) )  ->  z  e.  suc  A )
1918un2122 36591 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2019ex 432 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2120alrimivv 1741 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
22 dftr2 4490 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2322biimpri 206 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
2421, 23syl 17 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
2524idiALT 36216 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 974   A.wal 1403    = wceq 1405    e. wcel 1842   Tr wtr 4488   suc csuc 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-un 3418  df-in 3420  df-ss 3427  df-sn 3972  df-uni 4191  df-tr 4489  df-suc 5415
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator