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Theorem iin0 4626
 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 4340 . 2
2 0ex 4582 . . . . . 6
3 n0i 3789 . . . . . 6
42, 3ax-mp 5 . . . . 5
5 0iin 4388 . . . . . 6
65eqeq1i 2464 . . . . 5
74, 6mtbir 299 . . . 4
8 iineq1 4345 . . . . 5
98eqeq1d 2459 . . . 4
107, 9mtbiri 303 . . 3
1110necon2ai 2692 . 2
121, 11impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  e.wcel 1818  =/=wne 2652   cvv 3109   c0 3784  |^|_ciin 4331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-nul 4581 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-nul 3785  df-iin 4333
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