Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  iin0r Unicode version

Theorem iin0r 3922
 Description: If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
Assertion
Ref Expression
iin0r
Distinct variable group:   ,

Proof of Theorem iin0r
StepHypRef Expression
1 0ex 3884 . . . . 5
2 n0i 3229 . . . . 5
31, 2ax-mp 7 . . . 4
4 0iin 3715 . . . . 5
54eqeq1i 2047 . . . 4
63, 5mtbir 596 . . 3
7 iineq1 3671 . . . 4
87eqeq1d 2048 . . 3
96, 8mtbiri 600 . 2
109necon2ai 2259 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1243   wcel 1393   wne 2204  cvv 2557  c0 3224  ciin 3658 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225  df-iin 3660 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator