Theorem otelxp1 4322

Description: The first member of an ordered triple of classes in a cross product belongs to first cross 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 4320	. 2 ⊢ (⟨⟨A, B⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → ⟨A, B⟩ ∈ (𝑅 × 𝑆))
2		opelxp1 4320	. 2 ⊢ (⟨A, B⟩ ∈ (𝑅 × 𝑆) → A ∈ 𝑅)
3	1, 2	syl 14	1 ⊢ (⟨⟨A, B⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → A ∈ 𝑅)

Syntax hints:   → wi 4   ∈ wcel 1390  ⟨cop 3370   × cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294

This theorem is referenced by: (None)