Theorem otel3xp 5722

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 4637	. . . 4 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2		3simpa 1147	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
3		opelxp 5712	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
4	2, 3	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
5		simp3 1137	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍)
6	4, 5	opelxpd 5715	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
7	1, 6	eqeltrid 2836	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍))
8		eleq1 2820	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑋 × 𝑌) × 𝑍)))
9	7, 8	imbitrrid 245	. 2 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)))
10	9	imp 406	1 ⊢ ((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ⟨cop 4634  ⟨cotp 4636   × cxp 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-opab 5211  df-xp 5682

This theorem is referenced by: (None)