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Theorem iin0r 4063
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 4025 . . . . 5  |-  (/)  e.  _V
2 n0i 3338 . . . . 5  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
31, 2ax-mp 5 . . . 4  |-  -.  _V  =  (/)
4 0iin 3841 . . . . 5  |-  |^|_ x  e.  (/)  (/)  =  _V
54eqeq1i 2125 . . . 4  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
63, 5mtbir 645 . . 3  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
7 iineq1 3797 . . . 4  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
87eqeq1d 2126 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
96, 8mtbiri 649 . 2  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
109necon2ai 2339 1  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1316    e. wcel 1465    =/= wne 2285   _Vcvv 2660   (/)c0 3333   |^|_ciin 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-v 2662  df-dif 3043  df-nul 3334  df-iin 3786
This theorem is referenced by: (None)
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