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Theorem ot2elxp 33418
Description: Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
Assertion
Ref Expression
ot2elxp (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))

Proof of Theorem ot2elxp
StepHypRef Expression
1 opelxp 5601 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴𝐷𝐵𝐸))
21anbi1i 627 . 2 ((⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
3 opelxp 5601 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹))
4 df-3an 1091 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
52, 3, 43bitr4i 306 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089  wcel 2111  cop 4561   × cxp 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5130  df-xp 5571
This theorem is referenced by:  frpoins3xp3g  33551  xpord3lem  33558  xpord3ind  33563
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