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Theorem elopaelxp 5778
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 5302, ax-nul 5312, ax-pr 5438. (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 3482 . . . . . 6 𝑥 ∈ V
2 vex 3482 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 470 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5560 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 df-xp 5695 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
75, 6sseqtrri 4033 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V)
87sseli 3991 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  {copab 5210   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695
This theorem is referenced by:  bropaex12  5780  clwlkcompim  29813  linedegen  36125  opelopab3  37705
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