Theorem elopaelxp 5722

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 5243, ax-nul 5253, ax-pr 5379. (Revised by SN, 11-Dec-2024.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem elopaelxp

Step	Hyp	Ref	Expression
1		vex 3446	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3446	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 470	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5506	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5638	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 3985	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
8	7	sseli 3931	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3442  {copab 5162   × cxp 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-opab 5163  df-xp 5638

This theorem is referenced by:  bropaex12  5723  clwlkcompim  29865  linedegen  36359  opelopab3  37969