Theorem elxp2 5687

Description: Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.)

Assertion

Ref	Expression
elxp2	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

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

Proof of Theorem elxp2

Step	Hyp	Ref	Expression
1		ancom 465	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2	1	2exbii 1879	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3		elxp 5686	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4		r2ex 3202	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
5	2, 3, 4	3bitr4i 306	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  ⟨cop 4596   × cxp 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-un 3911  df-in 3913  df-ss 3923  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5175  df-xp 5669

This theorem is referenced by:  opelxp  5699  xpiundi  5734  xpiundir  5735  ssrel2  5773  reuop  6296  el2xptp  8033  f1o2ndf1  8118  frpoins3xpg  8137  poxp2  8140  xpord2pred  8142  sexp2  8143  xpdom2  9061  tskxpss  10758  nqereu  10915  elreal  11117  xpsmnd0  18837  efgmnvl  19785  frgpuptinv  19842  frgpup3lem  19848  xpsring1d  20416  pzriprnglem3  21614  pzriprnglem8  21619  pzriprnglem10  21621  ucnima  24418  ltgseg  28846  suppovss  33007  elrlocbasi  33568  qtophaus  34207  esum2dlem  34463  bj-mpomptALT  37742  fourierdlem42  46846  gpgvtxel  48795