Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  suctr Structured version   Unicode version

Theorem suctr 4667
 Description: The sucessor of a transitive class is transitive. The proof of http://www.virtualdeduction.com/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctr 4667 using completeusersproof, which is verified by the Metamath program. The proof of http://www.virtualdeduction.com/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 11-Apr-2009.) See suctrALT 4666 for the original proof before this revision. (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
suctr

Proof of Theorem suctr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 4660 . . . . . . 7
2 id 21 . . . . . . . 8
3 id 21 . . . . . . . . 9
43simpld 447 . . . . . . . 8
5 id 21 . . . . . . . 8
6 trel 4311 . . . . . . . . . 10
763impib 1152 . . . . . . . . 9
87idi 2 . . . . . . . 8
92, 4, 5, 8syl3an 1227 . . . . . . 7
101, 9sseldi 3348 . . . . . 6
11103expia 1156 . . . . 5
124adantr 453 . . . . . . . . 9
13 id 21 . . . . . . . . . 10
1413adantl 454 . . . . . . . . 9
1512, 14eleqtrd 2514 . . . . . . . 8
161, 15sseldi 3348 . . . . . . 7
1716ex 425 . . . . . 6
1817adantl 454 . . . . 5
193simprd 451 . . . . . . 7
20 elsuci 4649 . . . . . . 7
2119, 20syl 16 . . . . . 6
2221adantl 454 . . . . 5
2311, 18, 22mpjaod 372 . . . 4
2423ex 425 . . 3
2524alrimivv 1643 . 2
26 dftr2 4306 . . 3
2726biimpri 199 . 2
2825, 27syl 16 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 is referenced by:  dfon2lem3  25414  dfon2lem7  25418  dford3lem2  27100 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