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Theorem elxp 4389
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 4378 . . 3 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
21eleq2i 2120 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
3 elopab 4022 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
42, 3bitri 177 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  cop 3405  {copab 3844   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378
This theorem is referenced by:  elxp2  4390  0nelxp  4399  0nelelxp  4400  rabxp  4407  elxp3  4421  elvv  4429  elvvv  4430  0xp  4447  xpmlem  4771  elxp4  4835  elxp5  4836  dfco2a  4848  opabex3d  5775  opabex3  5776  xp1st  5819  xp2nd  5820  poxp  5880  xpsnen  6325  xpcomco  6330  xpassen  6334  nqnq0pi  6593
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