Theorem elopaelxp 5752

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 5261, ax-nul 5271, ax-pr 5405. (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 475	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5536	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5668	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 3994	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
8	7	sseli 3941	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  Vcvv 3463  {copab 5177   × cxp 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-opab 5178  df-xp 5668

This theorem is referenced by:  bropaex12  5753  clwlkcompim  30069  linedegen  36533  opelopab3  38256