Theorem trun 5227

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

Assertion

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

Proof of Theorem trun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4103	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		trss 5226	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	adantr 486	. . . . . 6 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
4		trss 5226	. . . . . . 7 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
5	4	adantl 487	. . . . . 6 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
6	3, 5	orim12d 979	. . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
7	1, 6	biimtrid 245	. . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
8		ssun 4144	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
9	7, 8	syl6 36	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵)))
10	9	ralrimiv 3155	. 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝑥 ⊆ (𝐴 ∪ 𝐵))
11		dftr3 5221	. 2 ⊢ (Tr (𝐴 ∪ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝑥 ⊆ (𝐴 ∪ 𝐵))
12	10, 11	sylibr 237	1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∪ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2145  ∀wral 3078   ∪ cun 3900   ⊆ wss 3902  Tr wtr 5216

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 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-un 3907  df-ss 3919  df-uni 4871  df-tr 5217

This theorem is used by: (None)