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Theorem elxp2 5122
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 466 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
212exbii 1773 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3 elxp 5121 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
4 r2ex 3057 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
52, 3, 43bitr4i 292 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wrex 2910  cop 4174   × cxp 5102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110
This theorem is referenced by:  opelxp  5136  xpiundi  5163  xpiundir  5164  ssrel2  5200  el2xptp  7196  f1o2ndf1  7270  xpdom2  8040  tskxpss  9579  nqereu  9736  elreal  9937  efgmnvl  18108  frgpuptinv  18165  frgpup3lem  18171  ucnima  22066  ltgseg  25472  qtophaus  29877  esum2dlem  30128  bj-mpt2mptALT  33047  fourierdlem42  40129
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