Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iun0 Structured version   Unicode version

Theorem iun0 4147
 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

Proof of Theorem iun0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 noel 3632 . . . . . 6
21a1i 11 . . . . 5
32nrex 2808 . . . 4
4 eliun 4097 . . . 4
53, 4mtbir 291 . . 3
65, 12false 340 . 2
76eqriv 2433 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1652   wcel 1725  wrex 2706  c0 3628  ciun 4093 This theorem is referenced by:  iununi  4175  funiunfv  5995  om0r  6783  kmlem11  8040  ituniiun  8302  voliunlem1  19444  sigaclfu2  24504  measvunilem0  24567  measvuni  24568  sibfof  24654  cvmscld  24960  trpred0  25514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-nul 3629  df-iun 4095
 Copyright terms: Public domain W3C validator