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Theorem trel3 3853
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
trel3 (Tr A → ((B 𝐶 𝐶 𝐷 𝐷 A) → B A))

Proof of Theorem trel3
StepHypRef Expression
1 3anass 888 . . 3 ((B 𝐶 𝐶 𝐷 𝐷 A) ↔ (B 𝐶 (𝐶 𝐷 𝐷 A)))
2 trel 3852 . . . 4 (Tr A → ((𝐶 𝐷 𝐷 A) → 𝐶 A))
32anim2d 320 . . 3 (Tr A → ((B 𝐶 (𝐶 𝐷 𝐷 A)) → (B 𝐶 𝐶 A)))
41, 3syl5bi 141 . 2 (Tr A → ((B 𝐶 𝐶 𝐷 𝐷 A) → (B 𝐶 𝐶 A)))
5 trel 3852 . 2 (Tr A → ((B 𝐶 𝐶 A) → B A))
64, 5syld 40 1 (Tr A → ((B 𝐶 𝐶 𝐷 𝐷 A) → B A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   wcel 1390  Tr wtr 3845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by: (None)
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