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

Theorem otel3xp 5151
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 4184 . . . 4 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 3simpa 1057 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴𝑋𝐵𝑌))
3 opelxp 5144 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴𝑋𝐵𝑌))
42, 3sylibr 224 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5 simp3 1062 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
64, 5opelxpd 5147 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
71, 6syl5eqel 2704 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8 eleq1 2688 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
97, 8syl5ibr 236 . 2 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
109imp 445 1 ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  cop 4181  cotp 4183   × cxp 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-ot 4184  df-opab 4711  df-xp 5118
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator