Theorem elon2

Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004)

	Ref	Expression
Assertion	elon2	⊢ ( 𝐴 ∈ On ↔ ( Ord 𝐴 ∧ 𝐴 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ On → 𝐴 ∈ V )
2		elong	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴 ) )
3	1 2	biadanii	⊢ ( 𝐴 ∈ On ↔ ( 𝐴 ∈ V ∧ Ord 𝐴 ) )
4	3	biancomi	⊢ ( 𝐴 ∈ On ↔ ( Ord 𝐴 ∧ 𝐴 ∈ V ) )