Theorem trun 5203

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 4093	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		trss 5202	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	adantr 480	. . . . . 6 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
4		trss 5202	. . . . . . 7 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
5	4	adantl 481	. . . . . 6 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
6	3, 5	orim12d 967	. . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
7	1, 6	biimtrid 242	. . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝐵) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
8		ssun 4135	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
9	7, 8	syl6 35	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵)))
10	9	ralrimiv 3128	. 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝑥 ⊆ (𝐴 ∪ 𝐵))
11		dftr3 5197	. 2 ⊢ (Tr (𝐴 ∪ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝑥 ⊆ (𝐴 ∪ 𝐵))
12	10, 11	sylibr 234	1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2114  ∀wral 3051   ∪ cun 3887   ⊆ wss 3889  Tr wtr 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-un 3894  df-ss 3906  df-uni 4851  df-tr 5193

This theorem is referenced by: (None)