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Theorem ixp0 7095
 Description: The infinite Cartesian product of a family with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8363. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixp0

Proof of Theorem ixp0
StepHypRef Expression
1 nne 2605 . . . 4
21rexbii 2730 . . 3
3 rexnal 2716 . . 3
42, 3bitr3i 243 . 2
5 ixpn0 7094 . . 3
65necon1bi 2647 . 2
74, 6sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1652   wne 2599  wral 2705  wrex 2706  c0 3628  cixp 7063 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-nul 3629  df-ixp 7064
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