Theorem 0un 4353

Description: The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0un	⊢ (∅ ∪ 𝐴) = 𝐴

Proof of Theorem 0un

Step	Hyp	Ref	Expression
1		uncom 4114	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4351	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2788	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1563   ∪ cun 3905  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-un 3912  df-nul 4289

This theorem is referenced by:  sspr  4796  sstp  4797  symdifv  5048  iunxdif3  5057  nlim2  8463  indconst0  12221  pwmndid  18988  pwmnd  18989  psdmullem  22288  ltslpss  28059  leslss  28060  mulsrid  28264  mulsproplem5  28271  mulsproplem6  28272  mulsproplem7  28273  mulsproplem8  28274  coprprop  32956  fzodif1  33049  cycpmrn  33376  dflringlem3  33703  dflring4  33705  bj-pr22val  37516  bj-snfromadj  37541  tfsconcat0i  43934  fiiuncl  45643  founiiun0  45766  infxrpnf  46018  prsal  46890  meadjun  47034  caragenuncllem  47084  carageniuncllem1  47093  hoidmvle  47172  iscnrm3rlem1  49569