Theorem 0un 4390

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 4150	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4388	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2754	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1534   ∪ cun 3944  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-dif 3949  df-un 3951  df-nul 4323

This theorem is referenced by:  nlim2  8512  pwmndid  18921  pwmnd  18922  psdmullem  22155  sltlpss  27927  slelss  27928  mulsrid  28111  mulsproplem5  28118  mulsproplem6  28119  mulsproplem7  28120  mulsproplem8  28121  coprprop  32611  fzodif1  32698  cycpmrn  33025  bj-pr22val  36739  bj-snfromadj  36764  metakunt17  41929  tfsconcat0i  43048  fiiuncl  44703  founiiun0  44833  infxrpnf  45097  prsal  45975  meadjun  46119  caragenuncllem  46169  carageniuncllem1  46178  hoidmvle  46257  iscnrm3rlem1  48310