Theorem oneluni

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

	Ref	Expression
Hypothesis	on.1	$$⊢A\in \mathrm{On}$$
Assertion	oneluni	$$⊢B\in A\to A\cup B=A$$

Proof

Step	Hyp	Ref	Expression
1		on.1	$$⊢A\in \mathrm{On}$$
2	1	onelssi	$$⊢B\in A\to B\subseteq A$$
3		ssequn2	$$⊢B\subseteq A\leftrightarrow A\cup B=A$$
4	2 3	sylib	$$⊢B\in A\to A\cup B=A$$