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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
Distinct variable group:   ,

Proof of Theorem iin0
StepHypRef Expression
1 iinconst 4245 . 2
2 0ex 4492 . . . . . 6
3 n0i 3702 . . . . . 6
42, 3ax-mp 5 . . . . 5
5 0iin 4293 . . . . . 6
65eqeq1i 2427 . . . . 5
74, 6mtbir 300 . . . 4
8 iineq1 4250 . . . . 5
98eqeq1d 2424 . . . 4
107, 9mtbiri 304 . . 3
1110necon2ai 2624 . 2
121, 11impbii 190 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 187   wceq 1437   wcel 1872   wne 2593  cvv 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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