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Metamath Proof Explorer

Theorem onelon

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. (Contributed by NM, 26-Oct-2003)

		Ref	Expression
	Assertion	onelon	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		ordelon	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On )