Theorem 0un 4393

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 4154	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4391	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2758	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1539   ∪ cun 3947  ∅c0 4323

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3952  df-un 3954  df-nul 4324

This theorem is referenced by:  nlim2  8494  pwmndid  18855  pwmnd  18856  sltlpss  27636  slelss  27637  mulsrid  27806  mulsproplem5  27813  mulsproplem6  27814  mulsproplem7  27815  mulsproplem8  27816  coprprop  32186  fzodif1  32269  cycpmrn  32570  bj-pr22val  36205  bj-snfromadj  36230  metakunt17  41309  tfsconcat0i  42399  fiiuncl  44055  founiiun0  44189  infxrpnf  44456  prsal  45334  meadjun  45478  caragenuncllem  45528  carageniuncllem1  45537  hoidmvle  45616  iscnrm3rlem1  47662