Theorem elopaelxp 5775

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 5296, ax-nul 5306, ax-pr 5432. (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 3484	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3484	. . . . . 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 5555	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5691	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 4033	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
8	7	sseli 3979	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3480  {copab 5205   × cxp 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691

This theorem is referenced by:  bropaex12  5777  clwlkcompim  29800  linedegen  36144  opelopab3  37725