Theorem elopaelxp 5749

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 5271, ax-nul 5281, ax-pr 5407. (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 3468	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3468	. . . . . 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 5530	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5665	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 4013	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
8	7	sseli 3959	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3464  {copab 5186   × cxp 5657

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-opab 5187  df-xp 5665

This theorem is referenced by:  bropaex12  5751  clwlkcompim  29767  linedegen  36166  opelopab3  37747