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Theorem elxp2 5690
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) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp2
StepHypRef Expression
1 ancom 460 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
212exbii 1843 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3 elxp 5689 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
4 r2ex 3187 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
52, 3, 43bitr4i 303 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wrex 3062  cop 4626   × cxp 5664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-opab 5201  df-xp 5672
This theorem is referenced by:  opelxp  5702  xpiundi  5736  xpiundir  5737  ssrel2  5775  reuop  6282  el2xptp  8014  f1o2ndf1  8102  frpoins3xpg  8120  poxp2  8123  xpord2pred  8125  sexp2  8126  xpdom2  9063  tskxpss  10763  nqereu  10920  elreal  11122  xpsmnd0  18698  efgmnvl  19624  frgpuptinv  19681  frgpup3lem  19687  xpsring1d  20222  pzriprnglem3  21338  pzriprnglem8  21343  pzriprnglem10  21345  ucnima  24108  ltgseg  28316  suppovss  32375  qtophaus  33305  esum2dlem  33579  bj-mpomptALT  36490  fourierdlem42  45350
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