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Theorem otelxp 5722
Description: Ordered triple membership in a triple Cartesian product. (Contributed by Scott Fenton, 31-Jan-2025.)
Assertion
Ref Expression
otelxp (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))

Proof of Theorem otelxp
StepHypRef Expression
1 opelxp 5714 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹))
2 opelxp 5714 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴𝐷𝐵𝐸))
31, 2bianbi 626 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
4 df-ot 4638 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
54eleq1i 2820 . 2 (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹))
6 df-3an 1087 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
73, 5, 63bitr4i 303 1 (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085  wcel 2099  cop 4635  cotp 4637   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-opab 5211  df-xp 5684
This theorem is referenced by:  frpoins3xp3g  8146  xpord3lem  8154  xpord3pred  8157  xpord3inddlem  8159
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