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Theorem ixp0 7522
 Description: The infinite Cartesian product of a family (x) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8884. (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 2658 . . . 4
21rexbii 2959 . . 3
3 rexnal 2905 . . 3
42, 3bitr3i 251 . 2
5 ixpn0 7521 . . 3
65necon1bi 2690 . 2
74, 6sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  =/=wne 2652  A.wral 2807  E.wrex 2808   c0 3784  X_cixp 7489 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-ixp 7490
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