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Theorem trel3 3890
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 900 . . 3 ((𝐵𝐶𝐶𝐷𝐷𝐴) ↔ (𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)))
2 trel 3889 . . . 4 (Tr 𝐴 → ((𝐶𝐷𝐷𝐴) → 𝐶𝐴))
32anim2d 324 . . 3 (Tr 𝐴 → ((𝐵𝐶 ∧ (𝐶𝐷𝐷𝐴)) → (𝐵𝐶𝐶𝐴)))
41, 3syl5bi 145 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → (𝐵𝐶𝐶𝐴)))
5 trel 3889 . 2 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
64, 5syld 44 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wcel 1409  Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883
This theorem is referenced by: (None)
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