Theorem trelded 45315

Description: Deduction form of trel 5231. 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 5231	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
5	4	3impib 1134	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)
6	1, 2, 3, 5	syl3an 1178	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2146  Tr wtr 5223

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-ss 3925  df-uni 4878  df-tr 5224

This theorem is used by:  suctrALT3  45673