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Theorem xpm 4805
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 4804 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2 eleq1 2145 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
32cbvexv 1838 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
4 eleq1 2145 . . . 4 (𝑏 = 𝑦 → (𝑏𝐵𝑦𝐵))
54cbvexv 1838 . . 3 (∃𝑏 𝑏𝐵 ↔ ∃𝑦 𝑦𝐵)
63, 5anbi12i 448 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
7 eleq1 2145 . . 3 (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
87cbvexv 1838 . 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 1422  wcel 1434   × cxp 4397
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4405
This theorem is referenced by:  ssxpbm  4818  xp11m  4821  xpexr2m  4824  unixpm  4918
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