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Theorem elopaelxp 5757
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 5420. (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
StepHypRef Expression
1 vex 3477 . . . . . 6 𝑥 ∈ V
2 vex 3477 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 471 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5543 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 df-xp 5675 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
75, 6sseqtrri 4015 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
87sseli 3974 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3473  {copab 5203   × cxp 5667
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-opab 5204  df-xp 5675
This theorem is referenced by:  bropaex12  5759  clwlkcompim  28902  linedegen  34943  opelopab3  36388
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