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Theorem otel3xp 5592
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 4566 . . . 4 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 3simpa 1140 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴𝑋𝐵𝑌))
3 opelxp 5584 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴𝑋𝐵𝑌))
42, 3sylibr 235 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5 simp3 1130 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
64, 5opelxpd 5586 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
71, 6eqeltrid 2914 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8 eleq1 2897 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
97, 8syl5ibr 247 . 2 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
109imp 407 1 ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cop 4563  cotp 4565   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-opab 5120  df-xp 5554
This theorem is referenced by: (None)
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