Theorem oion

Description: The order type of the well-order R on A is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015) (Revised by Mario Carneiro, 23-May-2015)

	Ref	Expression
Hypothesis	oicl.1	⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 )
Assertion	oion	⊢ ( 𝐴 ∈ 𝑉 → dom 𝐹 ∈ On )

Proof

Step	Hyp	Ref	Expression
1		oicl.1	⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 )
2	1	oicl	⊢ Ord dom 𝐹
3	1	oiexg	⊢ ( 𝐴 ∈ 𝑉 → 𝐹 ∈ V )
4		dmexg	⊢ ( 𝐹 ∈ V → dom 𝐹 ∈ V )
5		elong	⊢ ( dom 𝐹 ∈ V → ( dom 𝐹 ∈ On ↔ Ord dom 𝐹 ) )
6	3 4 5	3syl	⊢ ( 𝐴 ∈ 𝑉 → ( dom 𝐹 ∈ On ↔ Ord dom 𝐹 ) )
7	2 6	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → dom 𝐹 ∈ On )