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Theorem ixp0 7495
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 8854. (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 2655 . . . 4  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
21rexbii 2956 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  E. x  e.  A  B  =  (/) )
3 rexnal 2902 . . 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 7494 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
65necon1bi 2687 . 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 1398    =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   X_cixp 7462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-nul 3784  df-ixp 7463
This theorem is referenced by: (None)
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