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Theorem suctrALT3 29016
Description: The successor of a transtive class is transitive. suctrALT3 29016 is the completed proof in conventional notation of the Virtual Deduction proof http://www.virtualdeduction.com/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 28636 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 28631). 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 4131) . (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 4485 . . . . . . . . 9  |-  A  C_  suc  A
2 id 19 . . . . . . . . . 10  |-  ( Tr  A  ->  Tr  A
)
3 id 19 . . . . . . . . . . 11  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 445 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 19 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  A )
62, 4, 5trelded 28630 . . . . . . . . 9  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
71, 6sseldi 3191 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
873expia 1153 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
9 id 19 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
10 eleq2 2357 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1110biimpac 472 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
124, 9, 11syl2an 463 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
131, 12sseldi 3191 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1413ex 423 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
153simprd 449 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
16 elsuci 4474 . . . . . . . 8  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1715, 16syl 15 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
188, 14, 17jaoded 28631 . . . . . 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 28879 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2019ex 423 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2120alrimivv 1622 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
22 dftr2 4131 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2322biimpri 197 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
2421, 23syl 15 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
2524idi 2 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   Tr wtr 4129   suc csuc 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-uni 3844  df-tr 4130  df-suc 4414
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