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Theorem suctrALT3 28745
Description: The successor of a transtive class is transitive. suctrALT3 28745 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 28369 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 28364). 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 4264) . (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 4618 . . . . . . . . 9  |-  A  C_  suc  A
2 id 20 . . . . . . . . . 10  |-  ( Tr  A  ->  Tr  A
)
3 id 20 . . . . . . . . . . 11  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( z  e.  y  /\  y  e. 
suc  A ) )
43simpld 446 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  y )
5 id 20 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  A )
62, 4, 5trelded 28363 . . . . . . . . 9  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
71, 6sseldi 3306 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
873expia 1155 . . . . . . 7  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  ( y  e.  A  ->  z  e. 
suc  A ) )
9 id 20 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
10 eleq2 2465 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
z  e.  y  <->  z  e.  A ) )
1110biimpac 473 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  y  =  A )  ->  z  e.  A )
124, 9, 11syl2an 464 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  A )
131, 12sseldi 3306 . . . . . . . 8  |-  ( ( ( z  e.  y  /\  y  e.  suc  A )  /\  y  =  A )  ->  z  e.  suc  A )
1413ex 424 . . . . . . 7  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  ( y  =  A  ->  z  e.  suc  A ) )
153simprd 450 . . . . . . . 8  |-  ( ( z  e.  y  /\  y  e.  suc  A )  ->  y  e.  suc  A )
16 elsuci 4607 . . . . . . . 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 28364 . . . . . 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 28611 . . . . 5  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A ) )  ->  z  e.  suc  A )
2019ex 424 . . . 4  |-  ( Tr  A  ->  ( (
z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) )
2120alrimivv 1639 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
22 dftr2 4264 . . . 4  |-  ( Tr 
suc  A  <->  A. z A. y
( ( z  e.  y  /\  y  e. 
suc  A )  -> 
z  e.  suc  A
) )
2322biimpri 198 . . 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 358    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   Tr wtr 4262   suc csuc 4543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-in 3287  df-ss 3294  df-sn 3780  df-uni 3976  df-tr 4263  df-suc 4547
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