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Theorem ixp0 7983
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 9343. (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 2827 . . . 4 𝐵 ≠ ∅ ↔ 𝐵 = ∅)
21rexbii 3070 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ∃𝑥𝐴 𝐵 = ∅)
3 rexnal 3024 . . 3 (∃𝑥𝐴 ¬ 𝐵 ≠ ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
42, 3bitr3i 266 . 2 (∃𝑥𝐴 𝐵 = ∅ ↔ ¬ ∀𝑥𝐴 𝐵 ≠ ∅)
5 ixpn0 7982 . . 3 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
65necon1bi 2851 . 2 (¬ ∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 = ∅)
74, 6sylbi 207 1 (∃𝑥𝐴 𝐵 = ∅ → X𝑥𝐴 𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wne 2823  wral 2941  wrex 2942  c0 3948  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949  df-ixp 7951
This theorem is referenced by:  vonioo  41217  vonicc  41220
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