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	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		vex	$$⊢y\in \mathrm{V}$$
3	1 2	pm3.2i	$$⊢\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)$$
4	3	a1i	$$⊢\psi \to \left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)$$
5	4	ssopab2i	$$⊢\left\{⟨x,y⟩|\psi \right\}\subseteq \left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)\right\}$$
6		df-xp	$$⊢\mathrm{V}\times \mathrm{V}=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)\right\}$$
7	5 6	sseqtrri	$$⊢\left\{⟨x,y⟩|\psi \right\}\subseteq \mathrm{V}\times \mathrm{V}$$
8	7	sseli	$$⊢A\in \left\{⟨x,y⟩|\psi \right\}\to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$