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Theorem trss 3945
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 → (𝐵𝐴𝐵𝐴))

Proof of Theorem trss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2150 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2 sseq1 3047 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
31, 2imbi12d 232 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝑥𝐴) ↔ (𝐵𝐴𝐵𝐴)))
43imbi2d 228 . . 3 (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥𝐴𝑥𝐴)) ↔ (Tr 𝐴 → (𝐵𝐴𝐵𝐴))))
5 dftr3 3940 . . . 4 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
6 rsp 2423 . . . 4 (∀𝑥𝐴 𝑥𝐴 → (𝑥𝐴𝑥𝐴))
75, 6sylbi 119 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
84, 7vtoclg 2679 . 2 (𝐵𝐴 → (Tr 𝐴 → (𝐵𝐴𝐵𝐴)))
98pm2.43b 51 1 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  wral 2359  wss 2999  Tr wtr 3936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937
This theorem is referenced by:  trin  3946  triun  3949  trintssm  3952  tz7.2  4181  ordelss  4206  trsucss  4250  ordsucss  4321
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