Theorem otelxp 5733

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 5725	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹))
2		opelxp 5725	. . 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 4640	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
5	4	eleq1i 2830	. 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 2106  ⟨cop 4637  ⟨cotp 4639   × cxp 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-opab 5211  df-xp 5695

This theorem is referenced by:  frpoins3xp3g  8165  xpord3lem  8173  xpord3pred  8176  xpord3inddlem  8178