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Metamath Proof Explorer

Theorem opelvv

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypotheses	opelvv.1	⊢ 𝐴 ∈ V
	opelvv.2	⊢ 𝐵 ∈ V
Assertion	opelvv	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( V × V )

Proof

Step	Hyp	Ref	Expression
1		opelvv.1	⊢ 𝐴 ∈ V
2		opelvv.2	⊢ 𝐵 ∈ V
3		opelxpi	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( V × V ) )
4	1 2 3	mp2an	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( V × V )