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Theorem ixpn0 8482
Description: The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9893. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixpn0 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)

Proof of Theorem ixpn0
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0 4307 . 2 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 df-ixp 8450 . . . . 5 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
32abeq2i 2945 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
4 ne0i 4297 . . . . 5 ((𝑓𝑥) ∈ 𝐵𝐵 ≠ ∅)
54ralimi 3157 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
63, 5simplbiim 505 . . 3 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
76exlimiv 1922 . 2 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
81, 7sylbi 218 1 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  c0 4288   Fn wfn 6343  cfv 6348  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-dif 3936  df-nul 4289  df-ixp 8450
This theorem is referenced by:  ixp0  8483  ac9  9893  ac9s  9903
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