Theorem trelded 44585

Description: Deduction form of trel 5268. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
trelded.1	⊢ (𝜑 → Tr 𝐴)
trelded.2	⊢ (𝜓 → 𝐵 ∈ 𝐶)
trelded.3	⊢ (𝜒 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
trelded	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴)

Proof of Theorem trelded

Step	Hyp	Ref	Expression
1		trelded.1	. 2 ⊢ (𝜑 → Tr 𝐴)
2		trelded.2	. 2 ⊢ (𝜓 → 𝐵 ∈ 𝐶)
3		trelded.3	. 2 ⊢ (𝜒 → 𝐶 ∈ 𝐴)
4		trel 5268	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
5	4	3impib 1117	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)
6	1, 2, 3, 5	syl3an 1161	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  Tr wtr 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260

This theorem is referenced by:  suctrALT3  44944