Theorem ot2elxp 33707

Description: Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)

Assertion

Ref	Expression
ot2elxp	⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))

Proof of Theorem ot2elxp

Step	Hyp	Ref	Expression
1		opelxp 5627	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸))
2	1	anbi1i 623	. 2 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹))
3		opelxp 5627	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹))
4		df-3an 1087	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹))
5	2, 3, 4	3bitr4i 302	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2101  ⟨cop 4570   × cxp 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-opab 5140  df-xp 5597

This theorem is referenced by:  frpoins3xp3g  33816  xpord3lem  33823  xpord3ind  33828