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Theorem xpmlem 4744
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 1807 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
2 vex 2560 . . . . . 6 𝑥 ∈ V
3 vex 2560 . . . . . 6 𝑦 ∈ V
42, 3opex 3966 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2100 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 opelxp 4374 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
75, 6syl6bb 185 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
84, 7spcev 2647 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
98exlimivv 1776 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
101, 9sylbir 125 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11 elxp 4362 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
12 simpr 103 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐴𝑦𝐵))
13122eximi 1492 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1411, 13sylbi 114 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1514exlimiv 1489 . . 3 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1615, 1sylib 127 . 2 (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
1710, 16impbii 117 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  cop 3378   × cxp 4343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351
This theorem is referenced by:  xpm  4745
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