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Theorem ixpn0 7086
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 8355. (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 3629 . 2  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
2 df-ixp 7056 . . . . . 6  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
32abeq2i 2542 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
f `  x )  e.  B ) )
43simprbi 451 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( f `  x
)  e.  B )
5 ne0i 3626 . . . . 5  |-  ( ( f `  x )  e.  B  ->  B  =/=  (/) )
65ralimi 2773 . . . 4  |-  ( A. x  e.  A  (
f `  x )  e.  B  ->  A. x  e.  A  B  =/=  (/) )
74, 6syl 16 . . 3  |-  ( f  e.  X_ x  e.  A  B  ->  A. x  e.  A  B  =/=  (/) )
87exlimiv 1644 . 2  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  B  =/=  (/) )
91, 8sylbi 188 1  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   (/)c0 3620    Fn wfn 5441   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  ixp0  7087  ac9  8355  ac9s  8365
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621  df-ixp 7056
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