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Theorem elxpi 5710
Description: Membership in a Cartesian product. Uses fewer axioms than elxp 5711. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxpi (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxpi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2738 . . . . 5 (𝑧 = 𝐴 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝑦⟩))
21anbi1d 631 . . . 4 (𝑧 = 𝐴 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
322exbidv 1921 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
4 df-xp 5694 . . . 4 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
5 df-opab 5210 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))}
64, 5eqtri 2762 . . 3 (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))}
73, 6elab2g 3682 . 2 (𝐴 ∈ (𝐵 × 𝐶) → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶))))
87ibi 267 1 (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  {cab 2711  cop 4636  {copab 5209   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-opab 5210  df-xp 5694
This theorem is referenced by:  xpdifid  6189  opreuopreu  8057  djuunxp  9958  rngqiprngimfo  21328  fsumdvdsmul  27252  fmla0xp  35367  mnringmulrcld  44223  rrx2plordisom  48572
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