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Theorem ixpn0 8104
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 9495. (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 4072 . 2 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 df-ixp 8073 . . . . 5 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
32abeq2i 2871 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
4 ne0i 4062 . . . . 5 ((𝑓𝑥) ∈ 𝐵𝐵 ≠ ∅)
54ralimi 3088 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
63, 5simplbiim 661 . . 3 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
76exlimiv 2005 . 2 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
81, 7sylbi 207 1 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1851  wcel 2137  {cab 2744  wne 2930  wral 3048  c0 4056   Fn wfn 6042  cfv 6047  Xcixp 8072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-v 3340  df-dif 3716  df-nul 4057  df-ixp 8073
This theorem is referenced by:  ixp0  8105  ac9  9495  ac9s  9505
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