Theorem trel3 5227

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 1111	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)))
2		trel 5226	. . . 4 ⊢ (Tr 𝐴 → ((𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐶 ∈ 𝐴))
3	2	anim2d 623	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
4	1, 3	biimtrid 245	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
5		trel 5226	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
6	4, 5	syld 48	1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  Tr wtr 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-uni 4873  df-tr 5219

This theorem is referenced by:  ordelord  6382