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Theorem ixp0 7095
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 8363. (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 2605 . . . 4  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
21rexbii 2730 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  E. x  e.  A  B  =  (/) )
3 rexnal 2716 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
42, 3bitr3i 243 . 2  |-  ( E. x  e.  A  B  =  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
5 ixpn0 7094 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
65necon1bi 2647 . 2  |-  ( -. 
A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =  (/) )
74, 6sylbi 188 1  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    =/= wne 2599   A.wral 2705   E.wrex 2706   (/)c0 3628   X_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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