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Theorem suctrALT3 27713
Description: The successor of a transtive class is transitive. suctrALT3 27713 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 27353 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 27348). 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 4055) . (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
StepHypRef Expression
1 sssucid 4406 . . . . . . . . 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 27347 . . . . . . . . 9  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  A )
71, 6sseldi 3120 . . . . . . . 8  |-  ( ( Tr  A  /\  (
z  e.  y  /\  y  e.  suc  A )  /\  y  e.  A
)  ->  z  e.  suc  A )
873expia 1158 . . . . . . 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 2317 . . . . . . . . . . 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 3120 . . . . . . . 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 4395 . . . . . . . 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 27348 . . . . . 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 27578 . . . . 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 2014 . . 3  |-  ( Tr  A  ->  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  -> 
z  e.  suc  A
) )
22 dftr2 4055 . . . 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 17 . 2  |-  ( Tr  A  ->  Tr  suc  A
)
2524idi 2 1  |-  ( Tr  A  ->  Tr  suc  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   Tr wtr 4053   suc csuc 4331
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101  df-ss 3108  df-sn 3587  df-uni 3769  df-tr 4054  df-suc 4335
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