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Theorem otelxp 5728
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 5720 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶𝐹))
2 opelxp 5720 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴𝐷𝐵𝐸))
31, 2bianbi 627 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ((𝐴𝐷𝐵𝐸) ∧ 𝐶𝐹))
4 df-ot 4634 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
54eleq1i 2831 . 2 (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹))
6 df-3an 1088 . 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 206  wa 395  w3a 1086  wcel 2107  cop 4631  cotp 4633   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-ot 4634  df-opab 5205  df-xp 5690
This theorem is referenced by:  frpoins3xp3g  8167  xpord3lem  8175  xpord3pred  8178  xpord3inddlem  8180
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