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Theorem xpm 4772
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 4771 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2 eleq1 2116 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
32cbvexv 1811 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
4 eleq1 2116 . . . 4 (𝑏 = 𝑦 → (𝑏𝐵𝑦𝐵))
54cbvexv 1811 . . 3 (∃𝑏 𝑏𝐵 ↔ ∃𝑦 𝑦𝐵)
63, 5anbi12i 441 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
7 eleq1 2116 . . 3 (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
87cbvexv 1811 . 2 (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
91, 6, 83bitr3i 203 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  wcel 1409   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378
This theorem is referenced by:  ssxpbm  4783  xp11m  4786  xpexr2m  4789  unixpm  4880
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