Theorem elxp2 5675

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 466	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
2	1	2exbii 1882	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3		elxp 5674	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4		r2ex 3200	. 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⟩)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  ⟨cop 4590   × cxp 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5657

This theorem is used by:  opelxp  5687  xpiundi  5722  xpiundir  5723  ssrel2  5761  el2xptp  5820  reuop  6295  f1o2ndf1  8131  frpoins3xpg  8150  poxp2  8153  xpord2pred  8155  sexp2  8156  xpdom2  9084  tskxpss  10850  nqereu  11007  elreal  11209  xpsmnd0  18965  efgmnvl  19921  frgpuptinv  19978  frgpup3lem  19984  xpsring1d  20556  pzriprnglem3  21782  pzriprnglem8  21787  pzriprnglem10  21789  ucnima  24592  ltgseg  29052  suppovss  33267  elrlocbasi  33821  qtophaus  34461  esum2dlem  34717  bj-mpomptALT  38020  fourierdlem42  47128  gpgvtxel  49114