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Theorem iin0r 3963
Description: If an indexed intersection of the empty set is empty, the index set is non-empty. (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 3925 . . . . 5  |-  (/)  e.  _V
2 n0i 3272 . . . . 5  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
31, 2ax-mp 7 . . . 4  |-  -.  _V  =  (/)
4 0iin 3756 . . . . 5  |-  |^|_ x  e.  (/)  (/)  =  _V
54eqeq1i 2090 . . . 4  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
63, 5mtbir 629 . . 3  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
7 iineq1 3712 . . . 4  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
87eqeq1d 2091 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
96, 8mtbiri 633 . 2  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
109necon2ai 2303 1  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2249   _Vcvv 2610   (/)c0 3267   |^|_ciin 3699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3924
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2612  df-dif 2984  df-nul 3268  df-iin 3701
This theorem is referenced by: (None)
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