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Theorem ixp0 7504
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 8866. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixp0  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )

Proof of Theorem ixp0
StepHypRef Expression
1 nne 2644 . . . 4  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
21rexbii 2945 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  E. x  e.  A  B  =  (/) )
3 rexnal 2891 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
42, 3bitr3i 251 . 2  |-  ( E. x  e.  A  B  =  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
5 ixpn0 7503 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
65necon1bi 2676 . 2  |-  ( -. 
A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =  (/) )
74, 6sylbi 195 1  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1383    =/= wne 2638   A.wral 2793   E.wrex 2794   (/)c0 3770   X_cixp 7471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-v 3097  df-dif 3464  df-nul 3771  df-ixp 7472
This theorem is referenced by: (None)
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