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Theorem xpm 4840
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
Assertion
Ref Expression
xpm ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑧,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem xpm
Dummy variables 𝑎 𝑏 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmlem 4839 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2 eleq1 2150 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
32cbvexv 1843 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
4 eleq1 2150 . . . 4 (𝑏 = 𝑦 → (𝑏𝐵𝑦𝐵))
54cbvexv 1843 . . 3 (∃𝑏 𝑏𝐵 ↔ ∃𝑦 𝑦𝐵)
63, 5anbi12i 448 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
7 eleq1 2150 . . 3 (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
87cbvexv 1843 . 2 (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
91, 6, 83bitr3i 208 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1426  wcel 1438   × cxp 4426
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434
This theorem is referenced by:  ssxpbm  4853  xp11m  4856  xpexr2m  4859  unixpm  4953
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