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	$$⊢\phi \to A\in \mathrm{On}$$
	onelond.2	$$⊢\phi \to B\in A$$
Assertion	onelond	$$⊢\phi \to B\in \mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		onelond.1	$$⊢\phi \to A\in \mathrm{On}$$
2		onelond.2	$$⊢\phi \to B\in A$$
3		onelon	$$⊢\left(A\in \mathrm{On}\wedge B\in A\right)\to B\in \mathrm{On}$$
4	1 2 3	syl2anc	$$⊢\phi \to B\in \mathrm{On}$$