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Theorem iin0 4621
Description: An indexed intersection of the empty set, with a nonempty 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 4335 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  (/)  =  (/) )
2 0ex 4577 . . . . . 6  |-  (/)  e.  _V
3 n0i 3790 . . . . . 6  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
42, 3ax-mp 5 . . . . 5  |-  -.  _V  =  (/)
5 0iin 4383 . . . . . 6  |-  |^|_ x  e.  (/)  (/)  =  _V
65eqeq1i 2474 . . . . 5  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
74, 6mtbir 299 . . . 4  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
8 iineq1 4340 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
98eqeq1d 2469 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
107, 9mtbiri 303 . . 3  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
1110necon2ai 2702 . 2  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
121, 11impbii 188 1  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-dif 3479  df-nul 3786  df-iin 4328
This theorem is referenced by: (None)
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