Theorem 0un 4347

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 4109	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4345	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2752	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1540   ∪ cun 3901  ∅c0 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-dif 3906  df-un 3908  df-nul 4285

This theorem is referenced by:  nlim2  8408  pwmndid  18810  pwmnd  18811  psdmullem  22050  sltlpss  27822  slelss  27823  mulsrid  28021  mulsproplem5  28028  mulsproplem6  28029  mulsproplem7  28030  mulsproplem8  28031  coprprop  32641  fzodif1  32735  cycpmrn  33085  bj-pr22val  36993  bj-snfromadj  37018  tfsconcat0i  43318  fiiuncl  45043  founiiun0  45168  infxrpnf  45425  prsal  46299  meadjun  46443  caragenuncllem  46493  carageniuncllem1  46502  hoidmvle  46581  iscnrm3rlem1  48924