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Theorem trel 3852
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
trel (Tr A → ((B 𝐶 𝐶 A) → B A))

Proof of Theorem trel
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 3847 . 2 (Tr Ayx((y x x A) → y A))
2 eleq12 2099 . . . . . 6 ((y = B x = 𝐶) → (y xB 𝐶))
3 eleq1 2097 . . . . . . 7 (x = 𝐶 → (x A𝐶 A))
43adantl 262 . . . . . 6 ((y = B x = 𝐶) → (x A𝐶 A))
52, 4anbi12d 442 . . . . 5 ((y = B x = 𝐶) → ((y x x A) ↔ (B 𝐶 𝐶 A)))
6 eleq1 2097 . . . . . 6 (y = B → (y AB A))
76adantr 261 . . . . 5 ((y = B x = 𝐶) → (y AB A))
85, 7imbi12d 223 . . . 4 ((y = B x = 𝐶) → (((y x x A) → y A) ↔ ((B 𝐶 𝐶 A) → B A)))
98spc2gv 2637 . . 3 ((B 𝐶 𝐶 A) → (yx((y x x A) → y A) → ((B 𝐶 𝐶 A) → B A)))
109pm2.43b 46 . 2 (yx((y x x A) → y A) → ((B 𝐶 𝐶 A) → B A))
111, 10sylbi 114 1 (Tr A → ((B 𝐶 𝐶 A) → B A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   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-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:  trel3  3853  trintssm  3861  ordtr1  4091  suctr  4124  trsuc  4125
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