Theorem otelxp 5728

Description: Ordered triple membership in a triple Cartesian product. (Contributed by Scott Fenton, 31-Jan-2025.)

Assertion

Ref	Expression
otelxp	⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))

Proof of Theorem otelxp

Step	Hyp	Ref	Expression
1		opelxp 5720	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹))
2		opelxp 5720	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸))
3	1, 2	bianbi 627	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹))
4		df-ot 4634	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
5	4	eleq1i 2831	. 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹))
6		df-3an 1088	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹))
7	3, 5, 6	3bitr4i 303	1 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2107  ⟨cop 4631  ⟨cotp 4633   × cxp 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-ot 4634  df-opab 5205  df-xp 5690

This theorem is referenced by:  frpoins3xp3g  8167  xpord3lem  8175  xpord3pred  8178  xpord3inddlem  8180