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Theorem ixp0 6865
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 8126. (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 2463 . . . 4  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
21rexbii 2581 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  E. x  e.  A  B  =  (/) )
3 rexnal 2567 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
42, 3bitr3i 242 . 2  |-  ( E. x  e.  A  B  =  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
5 ixpn0 6864 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
65necon1bi 2502 . 2  |-  ( -. 
A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =  (/) )
74, 6sylbi 187 1  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468   X_cixp 6833
This theorem is referenced by:  ac9  8126  ac9s  8136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469  df-ixp 6834
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