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Theorem xpmlem 4820
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
Assertion
Ref Expression
xpmlem ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem xpmlem
StepHypRef Expression
1 eeanv 1852 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
2 vex 2618 . . . . . 6 𝑥 ∈ V
3 vex 2618 . . . . . 6 𝑦 ∈ V
42, 3opex 4032 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2147 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 opelxp 4442 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
75, 6syl6bb 194 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
84, 7spcev 2706 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
98exlimivv 1821 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
101, 9sylbir 133 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11 elxp 4430 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
12 simpr 108 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐴𝑦𝐵))
13122eximi 1535 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1411, 13sylbi 119 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1514exlimiv 1532 . . 3 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1615, 1sylib 120 . 2 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
1710, 16impbii 124 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  cop 3434   × cxp 4411
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 3934  ax-pow 3986  ax-pr 4012
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-ral 2360  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 3877  df-xp 4419
This theorem is referenced by:  xpm  4821
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