Theorem oneluni

Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994)

	Ref	Expression
Hypothesis	on.1	⊢ 𝐴 ∈ On
Assertion	oneluni	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐴 ∪ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2	1	onelssi	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 )
3		ssequn2	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐴 )
4	2 3	sylib	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐴 ∪ 𝐵 ) = 𝐴 )