Theorem trel3 5213

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 1105	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)))
2		trel 5212	. . . 4 ⊢ (Tr 𝐴 → ((𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐶 ∈ 𝐴))
3	2	anim2d 621	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
4	1, 3	biimtrid 244	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
5		trel 5212	. 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 399   ∧ w3a 1097   ∈ wcel 2141  Tr wtr 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3919  df-uni 4863  df-tr 5205

This theorem is referenced by:  ordelord  6362