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Theorem elxp2 5701
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 462 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
212exbii 1852 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3 elxp 5700 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
4 r2ex 3196 . 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 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  cop 4635   × cxp 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683
This theorem is referenced by:  opelxp  5713  xpiundi  5747  xpiundir  5748  ssrel2  5786  reuop  6293  el2xptp  8021  f1o2ndf1  8108  frpoins3xpg  8126  poxp2  8129  xpord2pred  8131  sexp2  8132  xpdom2  9067  tskxpss  10767  nqereu  10924  elreal  11126  xpsmnd0  18666  efgmnvl  19582  frgpuptinv  19639  frgpup3lem  19645  xpsring1d  20146  ucnima  23786  ltgseg  27847  suppovss  31906  qtophaus  32816  esum2dlem  33090  bj-mpomptALT  36000  fourierdlem42  44865  pzriprnglem3  46807  pzriprnglem8  46812  pzriprnglem10  46814
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