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Theorem ot2elxp 33707
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 5627 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴𝐷𝐵𝐸))
21anbi1i 623 . 2 ((⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
3 opelxp 5627 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹))
4 df-3an 1087 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
52, 3, 43bitr4i 302 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085  wcel 2101  cop 4570   × cxp 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-opab 5140  df-xp 5597
This theorem is referenced by:  frpoins3xp3g  33816  xpord3lem  33823  xpord3ind  33828
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