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Theorem suctr 5505
 Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
Assertion
Ref Expression
suctr

Proof of Theorem suctr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 463 . . . . 5
2 vex 3047 . . . . . 6
32elsuc 5491 . . . . 5
41, 3sylib 200 . . . 4
5 simpl 459 . . . . . . 7
6 eleq2 2517 . . . . . . 7
75, 6syl5ibcom 224 . . . . . 6
8 elelsuc 5494 . . . . . 6
97, 8syl6 34 . . . . 5
10 trel 4503 . . . . . . . . 9
1110expd 438 . . . . . . . 8
1211adantrd 470 . . . . . . 7
1312, 8syl8 72 . . . . . 6
14 jao 515 . . . . . 6
1513, 14syl6 34 . . . . 5
169, 15mpdi 43 . . . 4
174, 16mpdi 43 . . 3
1817alrimivv 1773 . 2
19 dftr2 4498 . 2
2018, 19sylibr 216 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 370   wa 371  wal 1441   wceq 1443   wcel 1886   wtr 4496   csuc 5424 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-v 3046  df-un 3408  df-in 3410  df-ss 3417  df-sn 3968  df-uni 4198  df-tr 4497  df-suc 5428 This theorem is referenced by:  dfon2lem3  30424  dfon2lem7  30428  dford3lem2  35876
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