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Theorem uni0 3870
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4165 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
Assertion
Ref Expression
uni0  |-  U. (/)  =  (/)

Proof of Theorem uni0
StepHypRef Expression
1 0ss 3496 . 2  |-  (/)  C_  { (/) }
2 uni0b 3868 . 2  |-  ( U. (/)  =  (/)  <->  (/)  C_  { (/) } )
31, 2mpbir 200 1  |-  U. (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843
This theorem is referenced by:  uniintsn  3915  iununi  4002  unizlim  4525  unisn2  4538  opswap  5175  unixp0  5222  unixpid  5223  iotanul  5250  funfv  5602  dffv2  5608  1stval  6140  2ndval  6141  1st0  6142  2nd0  6143  1st2val  6161  2nd2val  6162  brtpos0  6257  tpostpos  6270  nnunifi  7124  infeq5  7354  rankuni  7551  rankxplim3  7567  iunfictbso  7757  cflim2  7905  fin1a2lem11  8052  itunisuc  8061  itunitc  8063  ttukeylem4  8155  incexclem  12311  arwval  13891  dprdsn  15287  zrhval  16478  0opn  16666  indistopon  16754  mretopd  16845  hauscmplem  17149  cmpfi  17151  alexsublem  17754  alexsubALTlem2  17758  ptcmplem2  17763  lebnumlem3  18477  1stnpr  23260  2ndnpr  23261  prsiga  23507  unisnif  24535  limsucncmpi  24956  ovoliunnfl  25001  empos  25345  comppfsc  26410  stoweidlem35  27887  stoweidlem39  27891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-uni 3844
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