Theorem 0un 4349

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 4111	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4347	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2752	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1540   ∪ cun 3903  ∅c0 4286

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 3440  df-dif 3908  df-un 3910  df-nul 4287

This theorem is referenced by:  nlim2  8415  pwmndid  18828  pwmnd  18829  psdmullem  22068  sltlpss  27840  slelss  27841  mulsrid  28039  mulsproplem5  28046  mulsproplem6  28047  mulsproplem7  28048  mulsproplem8  28049  coprprop  32655  fzodif1  32748  cycpmrn  33098  bj-pr22val  36992  bj-snfromadj  37017  tfsconcat0i  43318  fiiuncl  45043  founiiun0  45168  infxrpnf  45426  prsal  46300  meadjun  46444  caragenuncllem  46494  carageniuncllem1  46503  hoidmvle  46582  iscnrm3rlem1  48925