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Theorem ixpn0 8749
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 10285. (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 4286 . 2 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 df-ixp 8717 . . . . 5 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
32abeq2i 2873 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
4 ne0i 4274 . . . . 5 ((𝑓𝑥) ∈ 𝐵𝐵 ≠ ∅)
54ralimi 3083 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
63, 5simplbiim 506 . . 3 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
76exlimiv 1931 . 2 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
81, 7sylbi 216 1 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1779  wcel 2104  {cab 2713  wne 2941  wral 3062  c0 4262   Fn wfn 6453  cfv 6458  Xcixp 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-dif 3895  df-nul 4263  df-ixp 8717
This theorem is referenced by:  ixp0  8750  ac9  10285  ac9s  10295
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