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Theorem trel 4037
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 4032 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 eleq12 2205 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝑥𝐵𝐶))
3 eleq1 2203 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
43adantl 275 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑥𝐴𝐶𝐴))
52, 4anbi12d 465 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → ((𝑦𝑥𝑥𝐴) ↔ (𝐵𝐶𝐶𝐴)))
6 eleq1 2203 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
76adantr 274 . . . . 5 ((𝑦 = 𝐵𝑥 = 𝐶) → (𝑦𝐴𝐵𝐴))
85, 7imbi12d 233 . . . 4 ((𝑦 = 𝐵𝑥 = 𝐶) → (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)))
98spc2gv 2777 . . 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 1330   = wceq 1332  wcel 1481  Tr wtr 4030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-in 3078  df-ss 3085  df-uni 3741  df-tr 4031
This theorem is referenced by:  trel3  4038  ordtr1  4314  suctr  4347  trsuc  4348  ordn2lp  4464
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