Theorem opex

Description: An ordered pair of classes is a set. Exercise 7 of TakeutiZaring p. 16. (Contributed by NM, 18-Aug-1993) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opex	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V

Proof

Step	Hyp	Ref	Expression
1		dfopif	⊢ ⟨ 𝐴 , 𝐵 ⟩ = if ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) , { { 𝐴 } , { 𝐴 , 𝐵 } } , ∅ )
2		prex	⊢ { { 𝐴 } , { 𝐴 , 𝐵 } } ∈ V
3		0ex	⊢ ∅ ∈ V
4	2 3	ifex	⊢ if ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) , { { 𝐴 } , { 𝐴 , 𝐵 } } , ∅ ) ∈ V
5	1 4	eqeltri	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V