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Theorem ixp0 6756
Description: The infinite Cartesian product of a family  B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
Assertion
Ref Expression
ixp0  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )

Proof of Theorem ixp0
Dummy variables  f  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 notm0 3458 . . . 4  |-  ( -. 
E. z  z  e.  B  <->  B  =  (/) )
21rexbii 2497 . . 3  |-  ( E. x  e.  A  -.  E. z  z  e.  B  <->  E. x  e.  A  B  =  (/) )
3 rexnalim 2479 . . 3  |-  ( E. x  e.  A  -.  E. z  z  e.  B  ->  -.  A. x  e.  A  E. z  z  e.  B )
42, 3sylbir 135 . 2  |-  ( E. x  e.  A  B  =  (/)  ->  -.  A. x  e.  A  E. z 
z  e.  B )
5 ixpm 6755 . . . 4  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
65con3i 633 . . 3  |-  ( -. 
A. x  e.  A  E. z  z  e.  B  ->  -.  E. f 
f  e.  X_ x  e.  A  B )
7 notm0 3458 . . 3  |-  ( -. 
E. f  f  e.  X_ x  e.  A  B 
<-> 
X_ x  e.  A  B  =  (/) )
86, 7sylib 122 . 2  |-  ( -. 
A. x  e.  A  E. z  z  e.  B  ->  X_ x  e.  A  B  =  (/) )
94, 8syl 14 1  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   (/)c0 3437   X_cixp 6723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-nul 3438  df-ixp 6724
This theorem is referenced by: (None)
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