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Theorem ixp0 8925
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 10478. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixp0 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)

Proof of Theorem ixp0
StepHypRef Expression
1 nne 2945 . . . 4 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
21rexbii 3095 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥𝐴 𝐵 = ∅)
3 rexnal 3101 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
42, 3bitr3i 277 . 2 (∃𝑥𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
5 ixpn0 8924 . . 3 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
65necon1bi 2970 . 2 (¬ ∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = ∅)
74, 6sylbi 216 1 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wne 2941  wral 3062  wrex 3071  c0 4323  Xcixp 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-dif 3952  df-nul 4324  df-ixp 8892
This theorem is referenced by:  vonioo  45398  vonicc  45401
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