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Theorem elxp2 5289
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 465 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
212exbii 1924 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) ↔ ∃𝑥𝑦((𝑥𝐵𝑦𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
3 elxp 5288 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
4 r2ex 3199 . 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 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  cop 4327   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272
This theorem is referenced by:  opelxp  5303  xpiundi  5330  xpiundir  5331  ssrel2  5367  el2xptp  7378  f1o2ndf1  7453  xpdom2  8220  tskxpss  9786  nqereu  9943  elreal  10144  efgmnvl  18327  frgpuptinv  18384  frgpup3lem  18390  ucnima  22286  ltgseg  25690  qtophaus  30212  esum2dlem  30463  bj-mpt2mptALT  33378  fourierdlem42  40869
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