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Theorem iin0r 4181
Description: If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
Assertion
Ref Expression
iin0r  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
Distinct variable group:    x, A

Proof of Theorem iin0r
StepHypRef Expression
1 0ex 4142 . . . . 5  |-  (/)  e.  _V
2 n0i 3440 . . . . 5  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
31, 2ax-mp 5 . . . 4  |-  -.  _V  =  (/)
4 0iin 3957 . . . . 5  |-  |^|_ x  e.  (/)  (/)  =  _V
54eqeq1i 2195 . . . 4  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
63, 5mtbir 672 . . 3  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
7 iineq1 3912 . . . 4  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
87eqeq1d 2196 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
96, 8mtbiri 676 . 2  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
109necon2ai 2411 1  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1363    e. wcel 2158    =/= wne 2357   _Vcvv 2749   (/)c0 3434   |^|_ciin 3899
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-nul 4141
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-v 2751  df-dif 3143  df-nul 3435  df-iin 3901
This theorem is referenced by: (None)
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