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Theorem trss 4005
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 2180 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2 sseq1 3090 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
31, 2imbi12d 233 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝑥𝐴) ↔ (𝐵𝐴𝐵𝐴)))
43imbi2d 229 . . 3 (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥𝐴𝑥𝐴)) ↔ (Tr 𝐴 → (𝐵𝐴𝐵𝐴))))
5 dftr3 4000 . . . 4 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
6 rsp 2457 . . . 4 (∀𝑥𝐴 𝑥𝐴 → (𝑥𝐴𝑥𝐴))
75, 6sylbi 120 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
84, 7vtoclg 2720 . 2 (𝐵𝐴 → (Tr 𝐴 → (𝐵𝐴𝐵𝐴)))
98pm2.43b 52 1 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  wral 2393  wss 3041  Tr wtr 3996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997
This theorem is referenced by:  trin  4006  triun  4009  trintssm  4012  tz7.2  4246  ordelss  4271  trsucss  4315  ordsucss  4390  ctinf  11870
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