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Theorem suctrALT3 29098
Description: The successor of a transtive class is transitive. suctrALT3 29098 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 28722 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 28715). 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 4306) . (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 4660 . . . . . . . . 9  |-  A  C_  suc  A
2 id 21 . . . . . . . . . 10  |-  ( Tr  A  ->  Tr  A
)
3 id 21 . . . . . . . . . . 11  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 447 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 21 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  A )
62, 4, 5trelded 28714 . . . . . . . . 9  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
71, 6sseldi 3348 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
873expia 1156 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
9 id 21 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
10 eleq2 2499 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1110biimpac 474 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
124, 9, 11syl2an 465 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
131, 12sseldi 3348 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1413ex 425 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
153simprd 451 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
16 elsuci 4649 . . . . . . . 8  |-  ( y  e.  suc  A  -> 
( y  e.  A  \/  y  =  A
) )
1715, 16syl 16 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  e.  A  \/  y  =  A ) )
188, 14, 17jaoded 28715 . . . . . 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 28964 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2019ex 425 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2120alrimivv 1643 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
22 dftr2 4306 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2322biimpri 199 . . 3  |-  ( A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A )  ->  Tr  suc  A
)
2421, 23syl 16 . 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 359    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   Tr wtr 4304   suc csuc 4585
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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