Theorem onelssi

Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994)

	Ref	Expression
Hypothesis	on.1	⊢ 𝐴 ∈ On
Assertion	onelssi	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2		onelss	⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) )
3	1 2	ax-mp	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 )