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Theorem iin0r 4053
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 4015 . . . . 5  |-  (/)  e.  _V
2 n0i 3334 . . . . 5  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
31, 2ax-mp 7 . . . 4  |-  -.  _V  =  (/)
4 0iin 3837 . . . . 5  |-  |^|_ x  e.  (/)  (/)  =  _V
54eqeq1i 2122 . . . 4  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
63, 5mtbir 643 . . 3  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
7 iineq1 3793 . . . 4  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
87eqeq1d 2123 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
96, 8mtbiri 647 . 2  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
109necon2ai 2336 1  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1314    e. wcel 1463    =/= wne 2282   _Vcvv 2657   (/)c0 3329   |^|_ciin 3780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-v 2659  df-dif 3039  df-nul 3330  df-iin 3782
This theorem is referenced by: (None)
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