Theorem onelond

Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of BellMachover p. 469. Lemma 1.3 of Schloeder p. 1. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		onelond.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		onelond.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3		onelon	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐵 ∈ On )