Theorem otelxp1 5694

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 5691	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
2		opelxp1 5691	. 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 2144  ⟨cop 4590   × cxp 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5165  df-xp 5655

This theorem is referenced by: (None)