Theorem elopaelxp

Description: Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018) Avoid ax-sep , ax-nul , ax-pr . (Revised by SN, 11-Dec-2024)

	Ref	Expression
Assertion	elopaelxp	⊢ ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → 𝐴 ∈ ( V × V ) )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2		vex	⊢ 𝑦 ∈ V
3	1 2	pm3.2i	⊢ ( 𝑥 ∈ V ∧ 𝑦 ∈ V )
4	3	a1i	⊢ ( 𝜓 → ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) )
5	4	ssopab2i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) }
6		df-xp	⊢ ( V × V ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) }
7	5 6	sseqtrri	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ⊆ ( V × V )
8	7	sseli	⊢ ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → 𝐴 ∈ ( V × V ) )