Theorem otelxp1 5745

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) → 𝐴 ∈ 𝑅)

Proof of Theorem otelxp1

Step	Hyp	Ref	Expression
1		opelxp1 5742	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
2		opelxp1 5742	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) → 𝐴 ∈ 𝑅)
3	1, 2	syl 17	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108  ⟨cop 4654   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706

This theorem is referenced by: (None)