Theorem ordelon

Description: An element of an ordinal class is an ordinal number. Lemma 1.3 of Schloeder p. 1. (Contributed by NM, 26-Oct-2003)

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

Proof

Step	Hyp	Ref	Expression
1		ordelord	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → Ord 𝐵 )
2		elong	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐵 ∈ On ↔ Ord 𝐵 ) )
3	2	adantl	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐵 ∈ On ↔ Ord 𝐵 ) )
4	1 3	mpbird	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ On )