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Theorem elxp2 4495
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 2381 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)))
2 r19.42v 2546 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3 an13 533 . . . . 5 ((𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
43exbii 1552 . . . 4 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
51, 2, 43bitr3i 209 . . 3 ((𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
65exbii 1552 . 2 (∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
7 df-rex 2381 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8 elxp 4494 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
96, 7, 83bitr4ri 212 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1299  wex 1436  wcel 1448  wrex 2376  cop 3477   × cxp 4475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-opab 3930  df-xp 4483
This theorem is referenced by:  opelxp  4507  xpiundi  4535  xpiundir  4536  ssrel2  4567  f1o2ndf1  6055  xpdom2  6654  elreal  7516
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