Theorem trun 5229

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 4107	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		trss 5228	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	adantr 485	. . . . . 6 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
4		trss 5228	. . . . . . 7 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
5	4	adantl 486	. . . . . 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 4148	. . . 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 3156	. 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝑥 ⊆ (𝐴 ∪ 𝐵))
11		dftr3 5223	. 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 400   ∨ wo 860   ∈ wcel 2143  ∀wral 3079   ∪ cun 3903   ⊆ wss 3905  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-un 3910  df-ss 3922  df-uni 4873  df-tr 5219

This theorem is used by: (None)