Theorem uni0 4943

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Contributed by NM, 16-Sep-1993.) Remove use of ax-nul 5311. (Revised by Eric Schmidt, 4-Apr-2007.)

Assertion

Ref	Expression
uni0	⊢ ∪ ∅ = ∅

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 4401	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 4941	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 230	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1534   ⊆ wss 3947  ∅c0 4325  {csn 4633  ∪ cuni 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-dif 3950  df-ss 3964  df-nul 4326  df-sn 4634  df-uni 4914

This theorem is referenced by:  csbuni  4944  uniintsn  4995  iununi  5107  unisn2  5317  eqsnuniex  5365  opswap  6240  unixp0  6294  unixpid  6295  unizlim  6499  iotanul  6532  funfv  6989  dffv2  6997  1stval  8005  2ndval  8006  1stnpr  8007  2ndnpr  8008  1st0  8009  2nd0  8010  1st2val  8031  2nd2val  8032  brtpos0  8248  tpostpos  8261  nnunifi  9328  supval2  9498  sup00  9507  infeq5  9680  rankuni  9906  rankxplim3  9924  iunfictbso  10157  cflim2  10306  fin1a2lem11  10453  itunisuc  10462  itunitc  10464  ttukeylem4  10555  relexpfldd  15055  incexclem  15840  arwval  18065  dprdsn  20036  zrhval  21497  0opn  22897  indistopon  22995  mretopd  23087  hauscmplem  23401  cmpfi  23403  comppfsc  23527  alexsublem  24039  alexsubALTlem2  24043  ptcmplem2  24048  lebnumlem3  24980  old0  27883  made0  27897  locfinref  33656  prsiga  33964  sigapildsys  33995  dya2iocuni  34117  fiunelcarsg  34150  carsgclctunlem1  34151  carsgclctunlem3  34154  wevgblacfn  34936  nnuni  35549  unisnif  35749  limsucncmpi  36157  heicant  37356  ovoliunnfl  37363  voliunnfl  37365  volsupnfl  37366  mbfresfi  37367  onov0suclim  42940  stoweidlem35  45656  stoweidlem39  45660  prsal  45939  issalnnd  45966  ismeannd  46088  caragenunicl  46145  isomennd  46152  ipolub0  48318