Theorem elxp 5603

Description: Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
elxp	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))

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

Proof of Theorem elxp

Step	Hyp	Ref	Expression
1		df-xp 5586	. . 3 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
3		elopab 5433	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	2, 3	bitri 274	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ⟨cop 4564  {copab 5132   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-xp 5586

This theorem is referenced by:  elxp2  5604  0nelxp  5614  0nelelxp  5615  rabxp  5626  elxp3  5644  elvv  5652  elvvv  5653  0xp  5675  dfres3  5885  xpdifid  6060  dfco2a  6139  elsnxp  6183  tpres  7058  elxp4  7743  elxp5  7744  opabex3d  7781  opabex3rd  7782  opabex3  7783  xp1st  7836  xp2nd  7837  poxp  7940  soxp  7941  xpsnen  8796  xpcomco  8802  xpassen  8806  dfac5lem1  9810  dfac5lem4  9813  axdc4lem  10142  fsum2dlem  15410  fprod2dlem  15618  numclwwlk1lem2fo  28623  satefvfmla0  33280  elima4  33656  brcart  34161  brimg  34166  dibelval3  39088