Theorem onsseli

Description: Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995)

	Ref	Expression
Hypotheses	on.1	$$⊢A\in \mathrm{On}$$
	on.2	$$⊢B\in \mathrm{On}$$
Assertion	onsseli	$$⊢A\subseteq B\leftrightarrow \left(A\in B\vee A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		on.1	$$⊢A\in \mathrm{On}$$
2		on.2	$$⊢B\in \mathrm{On}$$
3		onsseleq	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A\subseteq B\leftrightarrow \left(A\in B\vee A=B\right)\right)$$
4	1 2 3	mp2an	$$⊢A\subseteq B\leftrightarrow \left(A\in B\vee A=B\right)$$