Theorem elong

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994)

	Ref	Expression
Assertion	elong	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ordeq	⊢ ( 𝑦 = 𝑥 → ( Ord 𝑦 ↔ Ord 𝑥 ) )
2		ordeq	⊢ ( 𝑥 = 𝐴 → ( Ord 𝑥 ↔ Ord 𝐴 ) )
3		df-on	⊢ On = { 𝑦 ∣ Ord 𝑦 }
4	1 2 3	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴 ) )