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Theorem uni0 3752
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4046 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 3390 . 2  |-  (/)  C_  { (/) }
2 uni0b 3750 . 2  |-  ( U. (/)  =  (/)  <->  (/)  C_  { (/) } )
31, 2mpbir 202 1  |-  U. (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1619    C_ wss 3078   (/)c0 3362   {csn 3544   U.cuni 3727
This theorem is referenced by:  uniintsn  3797  iununi  3884  unizlim  4400  unisn2  4413  opswap  5065  unixp0  5112  unixpid  5113  fvprc  5374  funfv  5438  dffv2  5444  1stval  5976  2ndval  5977  1st0  5978  2nd0  5979  1st2val  5997  2nd2val  5998  brtpos0  6093  tpostpos  6106  iotanul  6158  nnunifi  6993  infeq5  7222  rankuni  7419  rankxplim3  7435  iunfictbso  7625  cflim2  7773  fin1a2lem11  7920  itunisuc  7929  itunitc  7931  ttukeylem4  8023  arwval  13719  dprdsn  15106  zrhval  16294  0opn  16482  indistopon  16570  mretopd  16661  hauscmplem  16965  cmpfi  16967  alexsublem  17570  alexsubALTlem2  17574  ptcmplem2  17579  lebnumlem3  18293  unisnif  23638  funpartfv  23657  limsucncmpi  24058  empos  24408  comppfsc  25473
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363  df-sn 3550  df-uni 3728
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