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Theorem trss 3863
 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 2100 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2 sseq1 2966 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
31, 2imbi12d 223 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝑥𝐴) ↔ (𝐵𝐴𝐵𝐴)))
43imbi2d 219 . . 3 (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥𝐴𝑥𝐴)) ↔ (Tr 𝐴 → (𝐵𝐴𝐵𝐴))))
5 dftr3 3858 . . . 4 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
6 rsp 2369 . . . 4 (∀𝑥𝐴 𝑥𝐴 → (𝑥𝐴𝑥𝐴))
75, 6sylbi 114 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
84, 7vtoclg 2613 . 2 (𝐵𝐴 → (Tr 𝐴 → (𝐵𝐴𝐵𝐴)))
98pm2.43b 46 1 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917  Tr wtr 3854 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855 This theorem is referenced by:  trin  3864  triun  3867  trint0m  3871  tz7.2  4091  ordelss  4116  trsucss  4160  ordsucss  4230
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