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Theorem trel3 3862
 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 889 . . 3 ((𝐵𝐶𝐶𝐷𝐷𝐴) ↔ (𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)))
2 trel 3861 . . . 4 (Tr 𝐴 → ((𝐶𝐷𝐷𝐴) → 𝐶𝐴))
32anim2d 320 . . 3 (Tr 𝐴 → ((𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)) → (𝐵𝐶𝐶𝐴)))
41, 3syl5bi 141 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → (𝐵𝐶𝐶𝐴)))
5 trel 3861 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
64, 5syld 40 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   ∈ wcel 1393  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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855 This theorem is referenced by: (None)
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