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Theorem ixpn0 7086
 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 8355. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixpn0

Proof of Theorem ixpn0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 n0 3629 . 2
2 df-ixp 7056 . . . . . 6
32abeq2i 2542 . . . . 5
43simprbi 451 . . . 4
5 ne0i 3626 . . . . 5
65ralimi 2773 . . . 4
74, 6syl 16 . . 3
87exlimiv 1644 . 2
91, 8sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wcel 1725  cab 2421   wne 2598  wral 2697  c0 3620   wfn 5441  cfv 5446  cixp 7055 This theorem is referenced by:  ixp0  7087  ac9  8355  ac9s  8365 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 2416 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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621  df-ixp 7056
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