Theorem trun

Description: The union of transitive classes is transitive. (Contributed by Eric Schmidt, 19-Apr-2026)

	Ref	Expression
Assertion	trun	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → Tr ( 𝐴 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
2		trss	⊢ ( Tr 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴 ) )
3	2	adantr	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴 ) )
4		trss	⊢ ( Tr 𝐵 → ( 𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵 ) )
5	4	adantl	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵 ) )
6	3 5	orim12d	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) → ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) ) )
7	1 6	biimtrid	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) ) )
8		ssun	⊢ ( ( 𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵 ) → 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) )
9	7 8	syl6	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) ) )
10	9	ralrimiv	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) )
11		dftr3	⊢ ( Tr ( 𝐴 ∪ 𝐵 ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) 𝑥 ⊆ ( 𝐴 ∪ 𝐵 ) )
12	10 11	sylibr	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → Tr ( 𝐴 ∪ 𝐵 ) )