Theorem onelssd

Description: An element of an ordinal number is a subset of the number. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026)

	Ref	Expression
Hypotheses	onelssd.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	onelssd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
Assertion	onelssd	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		onelssd.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		onelssd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3		onelss	⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) )
4	1 2 3	sylc	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )