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Theorem elopaelxp 5726
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 5268, ax-pr 5389. (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 3452 . . . . . 6 𝑥 ∈ V
2 vex 3452 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 472 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5512 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 df-xp 5644 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
75, 6sseqtrri 3986 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
87sseli 3945 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3448  {copab 5172   × cxp 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932  df-opab 5173  df-xp 5644
This theorem is referenced by:  bropaex12  5728  clwlkcompim  28770  linedegen  34757  opelopab3  36205
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