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Theorem trss 3854
Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
trss (Tr A → (B ABA))

Proof of Theorem trss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = B → (x AB A))
2 sseq1 2960 . . . . 5 (x = B → (xABA))
31, 2imbi12d 223 . . . 4 (x = B → ((x AxA) ↔ (B ABA)))
43imbi2d 219 . . 3 (x = B → ((Tr A → (x AxA)) ↔ (Tr A → (B ABA))))
5 dftr3 3849 . . . 4 (Tr Ax A xA)
6 rsp 2363 . . . 4 (x A xA → (x AxA))
75, 6sylbi 114 . . 3 (Tr A → (x AxA))
84, 7vtoclg 2607 . 2 (B A → (Tr A → (B ABA)))
98pm2.43b 46 1 (Tr A → (B ABA))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wral 2300  wss 2911  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  trin  3855  triun  3858  trint0m  3862  ordelss  4082  trsucss  4126  ordsucss  4196
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