Theorem el2xptp 7102

Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

Assertion

Ref	Expression
el2xptp	⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧

Proof of Theorem el2xptp

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp2 5056	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2		opeq1 4340	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
3	2	eqeq2d 2620	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
4	3	rexbidv 3034	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
5	4	rexxp 5186	. 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6		df-ot 4134	. . . . . . 7 ⊢ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
7	6	eqcomi 2619	. . . . . 6 ⊢ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧⟩
8	7	eqeq2i 2622	. . . . 5 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
9	8	rexbii 3023	. . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
10	9	rexbii 3023	. . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
11	10	rexbii 3023	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
12	1, 5, 11	3bitri 285	1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131  ⟨cotp 4133   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-iun 4457  df-opab 4644  df-xp 5044  df-rel 5045

This theorem is referenced by:  2spotmdisj  26595