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Theorem iin0 4265
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 3995 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  (/)  =  (/) )
2 0ex 4231 . . . . . 6  |-  (/)  e.  _V
3 n0i 3536 . . . . . 6  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
42, 3ax-mp 8 . . . . 5  |-  -.  _V  =  (/)
5 0iin 4041 . . . . . 6  |-  |^|_ x  e.  (/)  (/)  =  _V
65eqeq1i 2365 . . . . 5  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
74, 6mtbir 290 . . . 4  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
8 iineq1 4000 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
98eqeq1d 2366 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
107, 9mtbiri 294 . . 3  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
1110necon2ai 2566 . 2  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
121, 11impbii 180 1  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   (/)c0 3531   |^|_ciin 3987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-v 2866  df-dif 3231  df-nul 3532  df-iin 3989
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