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Theorem xpm 4960
 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 4959 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2 eleq1 2202 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
32cbvexv 1890 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
4 eleq1 2202 . . . 4 (𝑏 = 𝑦 → (𝑏𝐵𝑦𝐵))
54cbvexv 1890 . . 3 (∃𝑏 𝑏𝐵 ↔ ∃𝑦 𝑦𝐵)
63, 5anbi12i 455 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
7 eleq1 2202 . . 3 (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
87cbvexv 1890 . 2 (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
91, 6, 83bitr3i 209 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1468   ∈ wcel 1480   × cxp 4537 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545 This theorem is referenced by:  ssxpbm  4974  xp11m  4977  xpexr2m  4980  unixpm  5074
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