MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  otelxp1 Structured version   Visualization version   GIF version

Theorem otelxp1 5676
Description: The first member of an ordered triple of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
otelxp1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴𝑅)

Proof of Theorem otelxp1
StepHypRef Expression
1 opelxp1 5673 . 2 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
2 opelxp1 5673 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) → 𝐴𝑅)
31, 2syl 17 1 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cop 4591   × cxp 5630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167  df-xp 5638
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator