Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elxpxp Structured version   Visualization version   GIF version

Theorem elxpxp 33425
Description: Membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
Assertion
Ref Expression
elxpxp (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 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 elxpxp
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elxp2 5590 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2 opeq1 4799 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
32eqeq2d 2749 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
43rexbidv 3224 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54rexxp 5726 . 2 (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
61, 5bitri 278 1 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2111  wrex 3063  cop 4562   × cxp 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-iun 4921  df-opab 5131  df-xp 5572  df-rel 5573
This theorem is referenced by:  elxpxpss  33426  ralxp3f  33427  frpoins3xp3g  33554  poxp3  33562  xpord3pred  33564  sexp3  33565
  Copyright terms: Public domain W3C validator