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Theorem trss 3837
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 2082 . . . . 5 (x = B → (x AB A))
2 sseq1 2943 . . . . 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 3832 . . . 4 (Tr Ax A xA)
6 rsp 2347 . . . 4 (x A xA → (x AxA))
75, 6sylbi 114 . . 3 (Tr A → (x AxA))
84, 7vtoclg 2590 . 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 1228   wcel 1374  wral 2284  wss 2894  Tr wtr 3828
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829
This theorem is referenced by:  trin  3838  triun  3841  trint0m  3845  ordelss  4065  trsucss  4110  ordsucss  4180
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