Theorem trel3 5266

Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)

Assertion

Ref	Expression
trel3	⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Proof of Theorem trel3

Step	Hyp	Ref	Expression
1		3anass 1092	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)))
2		trel 5265	. . . 4 ⊢ (Tr 𝐴 → ((𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐶 ∈ 𝐴))
3	2	anim2d 611	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
4	1, 3	biimtrid 241	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
5		trel 5265	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
6	4, 5	syld 47	1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098  Tr wtr 5256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-uni 4901  df-tr 5257

This theorem is referenced by:  ordelord  6377