Theorem elxp2 5662

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 461	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2	1	2exbii 1851	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3		elxp 5661	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4		r2ex 3188	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
5	2, 3, 4	3bitr4i 302	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3069  ⟨cop 4597   × cxp 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644

This theorem is referenced by:  opelxp  5674  xpiundi  5707  xpiundir  5708  ssrel2  5746  reuop  6250  el2xptp  7972  f1o2ndf1  8059  frpoins3xpg  8077  poxp2  8080  xpord2pred  8082  sexp2  8083  xpdom2  9018  tskxpss  10717  nqereu  10874  elreal  11076  efgmnvl  19510  frgpuptinv  19567  frgpup3lem  19573  ucnima  23670  ltgseg  27601  suppovss  31665  qtophaus  32506  esum2dlem  32780  bj-mpomptALT  35663  fourierdlem42  44510