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Theorem trel 3973
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 3968 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 eleq12 2164 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝑥𝐵𝐶))
3 eleq1 2162 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
43adantl 273 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑥𝐴𝐶𝐴))
52, 4anbi12d 460 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → ((𝑦𝑥𝑥𝐴) ↔ (𝐵𝐶𝐶𝐴)))
6 eleq1 2162 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
76adantr 272 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝐴𝐵𝐴))
85, 7imbi12d 233 . . . 4 ((𝑦 = 𝐵𝑥 = 𝐶) → (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)))
98spc2gv 2731 . . 3 ((𝐵𝐶𝐶𝐴) → (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)))
109pm2.43b 52 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
111, 10sylbi 120 1 (Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1297   = wceq 1299  wcel 1448  Tr wtr 3966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-uni 3684  df-tr 3967
This theorem is referenced by:  trel3  3974  ordtr1  4248  suctr  4281  trsuc  4282  ordn2lp  4398
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