Theorem el2xptp 5819

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 5674	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2		opeq1 4839	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
3	2	eqeq2d 2771	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
4	3	rexbidv 3186	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
5	4	rexxp 5818	. 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6		df-ot 4858	. . . . . . 7 ⊢ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩
7	6	eqcomi 2769	. . . . . 6 ⊢ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧⟩
8	7	eqeq2i 2773	. . . . 5 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
9	8	rexbii 3109	. . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
10	9	rexbii 3109	. . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
11	10	rexbii 3109	. 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 300	1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ⟨cop 4834  ⟨cotp 4857   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-iun 4952  df-opab 5167  df-xp 5656  df-rel 5657

This theorem is used by:  mpt3mpt  7685  mpt3fvd  7688  el2xpss  8048  ralxp3f  8156  frpoins3xp3g  8160  poxp3  8169  xpord3pred  8171  sexp3  8172