Theorem uni0 4497

Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4822 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)

Assertion

Ref	Expression
uni0	⊢ ∪ ∅ = ∅

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 4005	. 2 ⊢ ∅ ⊆ {∅}
2		uni0b 4495	. 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅})
3	1, 2	mpbir 221	1 ⊢ ∪ ∅ = ∅

Syntax hints:   = wceq 1523   ⊆ wss 3607  ∅c0 3948  {csn 4210  ∪ cuni 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469

This theorem is referenced by:  csbuni  4498  uniintsn  4546  iununi  4642  unisn2  4827  opswap  5660  unixp0  5707  unixpid  5708  unizlim  5882  iotanul  5904  funfv  6304  dffv2  6310  1stval  7212  2ndval  7213  1stnpr  7214  2ndnpr  7215  1st0  7216  2nd0  7217  1st2val  7238  2nd2val  7239  brtpos0  7404  tpostpos  7417  nnunifi  8252  supval2  8402  sup00  8411  infeq5  8572  rankuni  8764  rankxplim3  8782  iunfictbso  8975  cflim2  9123  fin1a2lem11  9270  itunisuc  9279  itunitc  9281  ttukeylem4  9372  incexclem  14612  arwval  16740  dprdsn  18481  zrhval  19904  0opn  20757  indistopon  20853  mretopd  20944  hauscmplem  21257  cmpfi  21259  comppfsc  21383  alexsublem  21895  alexsubALTlem2  21899  ptcmplem2  21904  lebnumlem3  22809  locfinref  30036  prsiga  30322  sigapildsys  30353  dya2iocuni  30473  fiunelcarsg  30506  carsgclctunlem1  30507  carsgclctunlem3  30510  unisnif  32157  limsucncmpi  32569  heicant  33574  ovoliunnfl  33581  voliunnfl  33583  volsupnfl  33584  mbfresfi  33586  stoweidlem35  40570  stoweidlem39  40574  prsal  40856  issalnnd  40881  ismeannd  41002  caragenunicl  41059  isomennd  41066