MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  otel3xp Structured version   Visualization version   GIF version

Theorem otel3xp 5579
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 4535 . . . 4 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 3simpa 1149 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴𝑋𝐵𝑌))
3 opelxp 5571 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴𝑋𝐵𝑌))
42, 3sylibr 237 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5 simp3 1139 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
64, 5opelxpd 5573 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
71, 6eqeltrid 2838 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8 eleq1 2821 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
97, 8syl5ibr 249 . 2 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
109imp 410 1 ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  cop 4532  cotp 4534   × cxp 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-dif 3856  df-un 3858  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-ot 4535  df-opab 5103  df-xp 5541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator