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Theorem ixpn0 6864
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 Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixpn0  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )

Proof of Theorem ixpn0
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 n0 3477 . 2  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
2 df-ixp 6834 . . . . . 6  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
32abeq2i 2403 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
f `  x )  e.  B ) )
43simprbi 450 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( f `  x
)  e.  B )
5 ne0i 3474 . . . . 5  |-  ( ( f `  x )  e.  B  ->  B  =/=  (/) )
65ralimi 2631 . . . 4  |-  ( A. x  e.  A  (
f `  x )  e.  B  ->  A. x  e.  A  B  =/=  (/) )
74, 6syl 15 . . 3  |-  ( f  e.  X_ x  e.  A  B  ->  A. x  e.  A  B  =/=  (/) )
87exlimiv 1624 . 2  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  B  =/=  (/) )
91, 8sylbi 187 1  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   (/)c0 3468    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  ixp0  6865
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-v 2803  df-dif 3168  df-nul 3469  df-ixp 6834
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