Theorem elopaelxp 5759

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 5292, ax-nul 5299, ax-pr 5421. (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 3467	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3467	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 469	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5544	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5676	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 4009	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
8	7	sseli 3968	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  Vcvv 3463  {copab 5203   × cxp 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-in 3946  df-ss 3956  df-opab 5204  df-xp 5676

This theorem is referenced by:  bropaex12  5761  clwlkcompim  29610  linedegen  35768  opelopab3  37220