Theorem 0un 4392

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 4153	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4390	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2760	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1541   ∪ cun 3946  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-un 3953  df-nul 4323

This theorem is referenced by:  nlim2  8489  pwmndid  18816  pwmnd  18817  sltlpss  27398  mulsrid  27566  mulsproplem5  27573  mulsproplem6  27574  mulsproplem7  27575  mulsproplem8  27576  coprprop  31916  fzodif1  31999  cycpmrn  32297  bj-pr22val  35895  bj-snfromadj  35920  metakunt17  40996  tfsconcat0i  42085  fiiuncl  43742  founiiun0  43878  infxrpnf  44146  prsal  45024  meadjun  45168  caragenuncllem  45218  carageniuncllem1  45227  hoidmvle  45306  iscnrm3rlem1  47563