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Theorem suctrALT3 37294
 Description: The successor of a transtive class is transitive. suctrALT3 37294 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 36910 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 36903). 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 4520) . (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 5519 . . . . . . . . 9
2 id 22 . . . . . . . . . 10
3 id 22 . . . . . . . . . . 11
43simpld 460 . . . . . . . . . 10
5 id 22 . . . . . . . . . 10
62, 4, 5trelded 36902 . . . . . . . . 9
71, 6sseldi 3462 . . . . . . . 8
873expia 1207 . . . . . . 7
9 id 22 . . . . . . . . . 10
10 eleq2 2496 . . . . . . . . . . 11
1110biimpac 488 . . . . . . . . . 10
124, 9, 11syl2an 479 . . . . . . . . 9
131, 12sseldi 3462 . . . . . . . 8
1413ex 435 . . . . . . 7
153simprd 464 . . . . . . . 8
16 elsuci 5508 . . . . . . . 8
1715, 16syl 17 . . . . . . 7
188, 14, 17jaoded 36903 . . . . . 6
1918un2122 37150 . . . . 5
2019ex 435 . . . 4
2120alrimivv 1768 . . 3
22 dftr2 4520 . . . 4
2322biimpri 209 . . 3
2421, 23syl 17 . 2
2524idiALT 36802 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 369   wa 370   w3a 982  wal 1435   wceq 1437   wcel 1872   wtr 4518   csuc 5444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-uni 4220  df-tr 4519  df-suc 5448 This theorem is referenced by: (None)
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