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Theorem elxp 5041
Description: Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxp (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp
StepHypRef Expression
1 df-xp 5030 . . 3 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
21eleq2i 2675 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
3 elopab 4894 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
42, 3bitri 262 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975  cop 4126  {copab 4632   × cxp 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-opab 4634  df-xp 5030
This theorem is referenced by:  elxp2  5042  elxp2OLD  5043  0nelxp  5053  0nelxpOLD  5054  0nelelxp  5055  rabxp  5064  elxp3  5078  elvv  5086  elvvv  5087  0xp  5108  xpdifid  5463  dfco2a  5534  elsnxp  5576  elsnxpOLD  5577  tpres  6345  elxp4  6976  elxp5  6977  opabex3d  7010  opabex3  7011  xp1st  7062  xp2nd  7063  poxp  7149  soxp  7150  xpsnen  7902  xpcomco  7908  xpassen  7912  dfac5lem1  8802  dfac5lem4  8805  axdc4lem  9133  fsum2dlem  14285  fprod2dlem  14491  numclwlk1lem2fo  26384  dfres3  30704  elima4  30726  brcart  31011  brimg  31016  dibelval3  35253  av-numclwlk1lem2fo  41523
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