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 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Proof of Theorem trel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 3968	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
2		eleq12 2164	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝑥 ↔ 𝐵 ∈ 𝐶))
3		eleq1 2162	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	3	adantl 273	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	2, 4	anbi12d 460	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
6		eleq1 2162	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	adantr 272	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8	5, 7	imbi12d 233	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)))
9	8	spc2gv 2731	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)))
10	9	pm2.43b 52	. 2 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
11	1, 10	sylbi 120	1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))

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