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Theorem elxp2 4429
Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
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 df-rex 2361 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)))
2 r19.42v 2520 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3 an13 528 . . . . 5 ((𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
43exbii 1539 . . . 4 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
51, 2, 43bitr3i 208 . . 3 ((𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
65exbii 1539 . 2 (∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
7 df-rex 2361 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8 elxp 4428 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
96, 7, 83bitr4ri 211 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  wrex 2356  cop 3434   × cxp 4409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3875  df-xp 4417
This theorem is referenced by:  opelxp  4440  xpiundi  4464  xpiundir  4465  ssrel2  4496  f1o2ndf1  5950  xpdom2  6499  elreal  7310
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