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Theorem suctrALT4 29020
 Description: The sucessor of a transitive class is transitive. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://www.virtualdeduction.com/suctralt3vd.html. (Contributed by Alan Sare, 11-Sep-2016.)
Assertion
Ref Expression
suctrALT4

Proof of Theorem suctrALT4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . . 8
2 simpr 447 . . . . . . . 8
31, 2syl 15 . . . . . . 7
4 elsuci 4474 . . . . . . 7
53, 4syl 15 . . . . . 6
6 sssucid 4485 . . . . . . . 8
7 simpl 443 . . . . . . . . . 10
81, 7syl 15 . . . . . . . . 9
9 id 19 . . . . . . . . 9
10 eleq2 2357 . . . . . . . . . 10
1110biimpac 472 . . . . . . . . 9
128, 9, 11syl2an 463 . . . . . . . 8
13 ssel2 3188 . . . . . . . 8
146, 12, 13sylancr 644 . . . . . . 7
1514ex 423 . . . . . 6
16 id 19 . . . . . . . . 9
17 id 19 . . . . . . . . 9
18 trel 4136 . . . . . . . . . 10
19183impib 1149 . . . . . . . . 9
2016, 8, 17, 19syl3an 1224 . . . . . . . 8
216, 20, 13sylancr 644 . . . . . . 7
22213expia 1153 . . . . . 6
23 jao 498 . . . . . . 7
24233imp31 28632 . . . . . 6
255, 15, 22, 24eel2221 28781 . . . . 5
2625ex 423 . . . 4
2726alrimivv 1622 . . 3
28 dftr2 4131 . . . 4
2928biimpri 197 . . 3
3027, 29syl 15 . 2
3130idi 2 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 357   wa 358   w3a 934  wal 1530   wceq 1632   wcel 1696   wss 3165   wtr 4129   csuc 4410 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-uni 3844  df-tr 4130  df-suc 4414
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