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Theorem xpmlem 5019
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 1919 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
2 vex 2725 . . . . . 6 𝑥 ∈ V
3 vex 2725 . . . . . 6 𝑦 ∈ V
42, 3opex 4202 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2227 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 opelxp 4629 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
75, 6bitrdi 195 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
84, 7spcev 2817 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
98exlimivv 1883 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
101, 9sylbir 134 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11 elxp 4616 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
12 simpr 109 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐴𝑦𝐵))
13122eximi 1588 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1411, 13sylbi 120 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1514exlimiv 1585 . . 3 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1615, 1sylib 121 . 2 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
1710, 16impbii 125 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1342  wex 1479  wcel 2135  cop 3574   × cxp 4597
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-opab 4039  df-xp 4605
This theorem is referenced by:  xpm  5020
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