Theorem otel3xp 5696

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⟩ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))

Proof of Theorem otel3xp

Step	Hyp	Ref	Expression
1		df-ot 4592	. . . 4 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2		3simpa 1166	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
3		opelxp 5686	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
4	2, 3	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5		simp3 1156	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍)
6	4, 5	opelxpd 5689	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
7	1, 6	eqeltrid 2864	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8		eleq1 2848	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
9	7, 8	imbitrrid 249	. 2 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
10	9	imp 412	1 ⊢ ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ⟨cop 4589  ⟨cotp 4591   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-opab 5167  df-xp 5656

This theorem is used by: (None)