Theorem trel3

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

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

Proof

Step	Hyp	Ref	Expression
1		3anass	⊢ ( ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) ↔ ( 𝐵 ∈ 𝐶 ∧ ( 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) ) )
2		trel	⊢ ( Tr 𝐴 → ( ( 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) → 𝐶 ∈ 𝐴 ) )
3	2	anim2d	⊢ ( Tr 𝐴 → ( ( 𝐵 ∈ 𝐶 ∧ ( 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ) ) )
4	1 3	syl5bi	⊢ ( Tr 𝐴 → ( ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ) ) )
5		trel	⊢ ( Tr 𝐴 → ( ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) )
6	4 5	syld	⊢ ( Tr 𝐴 → ( ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) )