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Theorem iin0 4337
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 4066 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  (/)  =  (/) )
2 0ex 4303 . . . . . 6  |-  (/)  e.  _V
3 n0i 3597 . . . . . 6  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
42, 3ax-mp 8 . . . . 5  |-  -.  _V  =  (/)
5 0iin 4113 . . . . . 6  |-  |^|_ x  e.  (/)  (/)  =  _V
65eqeq1i 2415 . . . . 5  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
74, 6mtbir 291 . . . 4  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
8 iineq1 4071 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
98eqeq1d 2416 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
107, 9mtbiri 295 . . 3  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
1110necon2ai 2616 . 2  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
121, 11impbii 181 1  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920   (/)c0 3592   |^|_ciin 4058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-nul 4302
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-v 2922  df-dif 3287  df-nul 3593  df-iin 4060
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