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Theorem elxp3 4663
Description: Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
Assertion
Ref Expression
elxp3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp3
StepHypRef Expression
1 elxp 4626 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
2 eqcom 2172 . . . 4 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
3 opelxp 4639 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐵𝑦𝐶))
42, 3anbi12i 457 . . 3 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
542exbii 1599 . 2 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
61, 5bitr4i 186 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  cop 3584   × cxp 4607
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-xp 4615
This theorem is referenced by:  optocl  4685
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