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Theorem iin0 4534
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 4245 . 2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  (/)  =  (/) )
2 0ex 4492 . . . . . 6  |-  (/)  e.  _V
3 n0i 3702 . . . . . 6  |-  ( (/)  e.  _V  ->  -.  _V  =  (/) )
42, 3ax-mp 5 . . . . 5  |-  -.  _V  =  (/)
5 0iin 4293 . . . . . 6  |-  |^|_ x  e.  (/)  (/)  =  _V
65eqeq1i 2427 . . . . 5  |-  ( |^|_ x  e.  (/)  (/)  =  (/)  <->  _V  =  (/) )
74, 6mtbir 300 . . . 4  |-  -.  |^|_ x  e.  (/)  (/)  =  (/)
8 iineq1 4250 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  (/)  =  |^|_ x  e.  (/)  (/) )
98eqeq1d 2424 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  (/)  =  (/)  <->  |^|_ x  e.  (/)  (/)  =  (/) ) )
107, 9mtbiri 304 . . 3  |-  ( A  =  (/)  ->  -.  |^|_ x  e.  A  (/)  =  (/) )
1110necon2ai 2624 . 2  |-  ( |^|_ x  e.  A  (/)  =  (/)  ->  A  =/=  (/) )
121, 11impbii 190 1  |-  ( A  =/=  (/)  <->  |^|_ x  e.  A  (/)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    = wceq 1437    e. wcel 1872    =/= wne 2593   _Vcvv 3016   (/)c0 3697   |^|_ciin 4236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-nul 4491
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-v 3018  df-dif 3375  df-nul 3698  df-iin 4238
This theorem is referenced by: (None)
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