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Theorem elxp 4605
Description: Membership in a cross 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 4594 . . 3 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
21eleq2i 2224 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
3 elopab 4220 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
42, 3bitri 183 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1335  wex 1472  wcel 2128  cop 3564  {copab 4026   × cxp 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4028  df-xp 4594
This theorem is referenced by:  elxp2  4606  0nelxp  4616  0nelelxp  4617  rabxp  4625  elxp3  4642  elvv  4650  elvvv  4651  0xp  4668  xpmlem  5008  elxp4  5075  elxp5  5076  dfco2a  5088  opabex3d  6071  opabex3  6072  xp1st  6115  xp2nd  6116  poxp  6181  xpsnen  6768  xpcomco  6773  xpassen  6777  nqnq0pi  7360  fsum2dlemstep  11342  fprod2dlemstep  11530
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