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Theorem trel3 5269
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 1093 . . 3 ((𝐵𝐶𝐶𝐷𝐷𝐴) ↔ (𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)))
2 trel 5268 . . . 4 (Tr 𝐴 → ((𝐶𝐷𝐷𝐴) → 𝐶𝐴))
32anim2d 611 . . 3 (Tr 𝐴 → ((𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)) → (𝐵𝐶𝐶𝐴)))
41, 3biimtrid 241 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → (𝐵𝐶𝐶𝐴)))
5 trel 5268 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
64, 5syld 47 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085  wcel 2099  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-ss 3962  df-uni 4904  df-tr 5260
This theorem is referenced by:  ordelord  6385
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