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Theorem ixp0 8478
 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 9890. (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 3017 . . . 4 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
21rexbii 3241 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥𝐴 𝐵 = ∅)
3 rexnal 3232 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
42, 3bitr3i 280 . 2 (∃𝑥𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
5 ixpn0 8477 . . 3 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
65necon1bi 3041 . 2 (¬ ∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = ∅)
74, 6sylbi 220 1 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ≠ wne 3013  ∀wral 3132  ∃wrex 3133  ∅c0 4274  Xcixp 8444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-dif 3921  df-nul 4275  df-ixp 8445 This theorem is referenced by:  vonioo  43163  vonicc  43166
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