Theorem omun

Description: The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025)

	Ref	Expression
Assertion	omun	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∪ 𝐵 ) ∈ ω )

Proof

Step	Hyp	Ref	Expression
1		ssequn1	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 )
2		eleq1a	⊢ ( 𝐵 ∈ ω → ( ( 𝐴 ∪ 𝐵 ) = 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
3	2	adantl	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
4	1 3	biimtrid	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
5		ssequn2	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐴 )
6		eleq1a	⊢ ( 𝐴 ∈ ω → ( ( 𝐴 ∪ 𝐵 ) = 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
7	6	adantr	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
8	5 7	biimtrid	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) )
9		nnord	⊢ ( 𝐴 ∈ ω → Ord 𝐴 )
10		nnord	⊢ ( 𝐵 ∈ ω → Ord 𝐵 )
11		ordtri2or2	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) )
12	9 10 11	syl2an	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) )
13	4 8 12	mpjaod	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∪ 𝐵 ) ∈ ω )