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Theorem trel3 5231
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
trel3 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))

Proof of Theorem trel3
StepHypRef Expression
1 3anass 1096 . . 3 ((𝐵𝐶𝐶𝐷𝐷𝐴) ↔ (𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)))
2 trel 5230 . . . 4 (Tr 𝐴 → ((𝐶𝐷𝐷𝐴) → 𝐶𝐴))
32anim2d 613 . . 3 (Tr 𝐴 → ((𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)) → (𝐵𝐶𝐶𝐴)))
41, 3biimtrid 241 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → (𝐵𝐶𝐶𝐴)))
5 trel 5230 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
64, 5syld 47 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  Tr wtr 5221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-in 3916  df-ss 3926  df-uni 4865  df-tr 5222
This theorem is referenced by:  ordelord  6338
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