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Theorem xpmlem 4927
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 1882 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
2 vex 2661 . . . . . 6 𝑥 ∈ V
3 vex 2661 . . . . . 6 𝑦 ∈ V
42, 3opex 4119 . . . . 5 𝑥, 𝑦⟩ ∈ V
5 eleq1 2178 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6 opelxp 4537 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
75, 6syl6bb 195 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
84, 7spcev 2752 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
98exlimivv 1850 . . 3 (∃𝑥𝑦(𝑥𝐴𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
101, 9sylbir 134 . 2 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11 elxp 4524 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
12 simpr 109 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → (𝑥𝐴𝑦𝐵))
13122eximi 1563 . . . . 5 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1411, 13sylbi 120 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑥𝐴𝑦𝐵))
1514exlimiv 1560 . . 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 1314  wex 1451  wcel 1463  cop 3498   × cxp 4505
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513
This theorem is referenced by:  xpm  4928
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