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Theorem trel 3861
 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 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))

Proof of Theorem trel
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 3856 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 eleq12 2102 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝑥𝐵𝐶))
3 eleq1 2100 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
43adantl 262 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑥𝐴𝐶𝐴))
52, 4anbi12d 442 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → ((𝑦𝑥𝑥𝐴) ↔ (𝐵𝐶𝐶𝐴)))
6 eleq1 2100 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
76adantr 261 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝐴𝐵𝐴))
85, 7imbi12d 223 . . . 4 ((𝑦 = 𝐵𝑥 = 𝐶) → (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)))
98spc2gv 2643 . . 3 ((𝐵𝐶𝐶𝐴) → (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)))
109pm2.43b 46 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
111, 10sylbi 114 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ 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-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:  trel3  3862  trintssm  3870  ordtr1  4125  suctr  4158  trsuc  4159  ordn2lp  4269
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