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Theorem trel3 4951
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 1117 . . 3 ((𝐵𝐶𝐶𝐷𝐷𝐴) ↔ (𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)))
2 trel 4950 . . . 4 (Tr 𝐴 → ((𝐶𝐷𝐷𝐴) → 𝐶𝐴))
32anim2d 606 . . 3 (Tr 𝐴 → ((𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)) → (𝐵𝐶𝐶𝐴)))
41, 3syl5bi 234 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → (𝐵𝐶𝐶𝐴)))
5 trel 4950 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
64, 5syld 47 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wcel 2157  Tr wtr 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-in 3774  df-ss 3781  df-uni 4627  df-tr 4944
This theorem is referenced by:  ordelord  5961
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