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Theorem ixpn0 7826
 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 9188. (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 3890 . 2 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 df-ixp 7795 . . . . . 6 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
32abeq2i 2722 . . . . 5 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
43simprbi 479 . . . 4 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
5 ne0i 3880 . . . . 5 ((𝑓𝑥) ∈ 𝐵𝐵 ≠ ∅)
65ralimi 2936 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
74, 6syl 17 . . 3 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
87exlimiv 1845 . 2 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
91, 8sylbi 206 1 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∅c0 3874   Fn wfn 5799  ‘cfv 5804  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875  df-ixp 7795 This theorem is referenced by:  ixp0  7827  ac9  9188  ac9s  9198
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