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Theorem iin0 4376
Description: An indexed intersection of the empty set, with a non-empty index set, is empty. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
iin0  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Distinct variable group:    x, A

Proof of Theorem iin0
StepHypRef Expression
1 iinconst 4104 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  (/)  =  (/) )
2 0ex 4342 . . . . . 6  |-  (/)  e.  _V
3 n0i 3635 . . . . . 6  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
42, 3ax-mp 5 . . . . 5  |-  -.  _V  =  (/)
5 0iin 4151 . . . . . 6  |-  |^|_ x  e.  (/)  (/)  =  _V
65eqeq1i 2445 . . . . 5  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
74, 6mtbir 292 . . . 4  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
8 iineq1 4109 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
98eqeq1d 2446 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
107, 9mtbiri 296 . . 3  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
1110necon2ai 2651 . 2  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
121, 11impbii 182 1  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   |^|_ciin 4096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-nul 3631  df-iin 4098
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