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Theorem xpm 5046
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 5045 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2 eleq1 2240 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
32cbvexv 1918 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
4 eleq1 2240 . . . 4 (𝑏 = 𝑦 → (𝑏𝐵𝑦𝐵))
54cbvexv 1918 . . 3 (∃𝑏 𝑏𝐵 ↔ ∃𝑦 𝑦𝐵)
63, 5anbi12i 460 . 2 ((∃𝑎 𝑎𝐴 ∧ ∃𝑏 𝑏𝐵) ↔ (∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵))
7 eleq1 2240 . . 3 (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
87cbvexv 1918 . 2 (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
91, 6, 83bitr3i 210 1 ((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1492  wcel 2148   × cxp 4621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4629
This theorem is referenced by:  ssxpbm  5060  xp11m  5063  xpexr2m  5066  unixpm  5160
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