ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iun0 Unicode version

Theorem iun0 3771
Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iun0  |-  U_ x  e.  A  (/)  =  (/)

Proof of Theorem iun0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 noel 3279 . . . . . 6  |-  -.  y  e.  (/)
21a1i 9 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2461 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 3719 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 629 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 650 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2082 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1287    e. wcel 1436   E.wrex 2356   (/)c0 3275   U_ciun 3715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-nul 3276  df-iun 3717
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator