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Theorem ixpn0 8486
 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 9897. (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 4308 . 2 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
2 df-ixp 8454 . . . . 5 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
32abeq2i 2946 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
4 ne0i 4298 . . . . 5 ((𝑓𝑥) ∈ 𝐵𝐵 ≠ ∅)
54ralimi 3158 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
63, 5simplbiim 507 . . 3 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
76exlimiv 1925 . 2 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 𝐵 ≠ ∅)
81, 7sylbi 219 1 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1774   ∈ wcel 2108  {cab 2797   ≠ wne 3014  ∀wral 3136  ∅c0 4289   Fn wfn 6343  ‘cfv 6348  Xcixp 8453 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-dif 3937  df-nul 4290  df-ixp 8454 This theorem is referenced by:  ixp0  8487  ac9  9897  ac9s  9907
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