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Metamath Proof Explorer

Theorem onelss

Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994) (Proof shortened by Andrew Salmon, 25-Jul-2011)

		Ref	Expression
	Assertion	onelss	⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		ordelss	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ⊆ 𝐴 )
3	2	ex	⊢ ( Ord 𝐴 → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) )
4	1 3	syl	⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) )