Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suctrALT3 Structured version   Unicode version

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

Proof of Theorem suctrALT3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4660 . . . . . . . . 9
2 id 21 . . . . . . . . . 10
3 id 21 . . . . . . . . . . 11
43simpld 447 . . . . . . . . . 10
5 id 21 . . . . . . . . . 10
62, 4, 5trelded 28714 . . . . . . . . 9
71, 6sseldi 3348 . . . . . . . 8
873expia 1156 . . . . . . 7
9 id 21 . . . . . . . . . 10
10 eleq2 2499 . . . . . . . . . . 11
1110biimpac 474 . . . . . . . . . 10
124, 9, 11syl2an 465 . . . . . . . . 9
131, 12sseldi 3348 . . . . . . . 8
1413ex 425 . . . . . . 7
153simprd 451 . . . . . . . 8
16 elsuci 4649 . . . . . . . 8
1715, 16syl 16 . . . . . . 7
188, 14, 17jaoded 28715 . . . . . 6
1918un2122 28964 . . . . 5
2019ex 425 . . . 4
2120alrimivv 1643 . . 3
22 dftr2 4306 . . . 4
2322biimpri 199 . . 3
2421, 23syl 16 . 2
2524idi 2 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 359   wa 360   w3a 937  wal 1550   wceq 1653   wcel 1726   wtr 4304   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
 Copyright terms: Public domain W3C validator