Theorem elxp 5055

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 5044	. . 3 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
3		elopab 4908	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	2, 3	bitri 263	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131  {copab 4642   × 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-v 3175  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-opab 4644  df-xp 5044

This theorem is referenced by:  elxp2  5056  elxp2OLD  5057  0nelxp  5067  0nelxpOLD  5068  0nelelxp  5069  rabxp  5078  elxp3  5092  elvv  5100  elvvv  5101  0xp  5122  xpdifid  5481  dfco2a  5552  elsnxp  5594  elsnxpOLD  5595  tpres  6371  elxp4  7003  elxp5  7004  opabex3d  7037  opabex3  7038  xp1st  7089  xp2nd  7090  poxp  7176  soxp  7177  xpsnen  7929  xpcomco  7935  xpassen  7939  dfac5lem1  8829  dfac5lem4  8832  axdc4lem  9160  fsum2dlem  14343  fprod2dlem  14549  numclwlk1lem2fo  26622  dfres3  30902  elima4  30924  brcart  31209  brimg  31214  dibelval3  35454  av-numclwlk1lem2fo  41525