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Theorem otel3xp 5077
 Description: An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
Assertion
Ref Expression
otel3xp ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))

Proof of Theorem otel3xp
StepHypRef Expression
1 df-ot 4134 . . . 4 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 3simpa 1051 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴𝑋𝐵𝑌))
3 opelxp 5070 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴𝑋𝐵𝑌))
42, 3sylibr 223 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5 simp3 1056 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
64, 5opelxpd 5073 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
71, 6syl5eqel 2692 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8 eleq1 2676 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
97, 8syl5ibr 235 . 2 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
109imp 444 1 ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ⟨cotp 4133   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-opab 4644  df-xp 5044 This theorem is referenced by:  el2wlkonot  26396  el2spthonot  26397  el2spthonot0  26398  el2wlksotot  26409  usg2spot2nb  26592  usgreg2spot  26594
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