Theorem 0un 4349

Description: The union of the empty set with a class is itself. Commuted form of un0 4347. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0un	⊢ (∅ ∪ 𝐴) = 𝐴

Proof of Theorem 0un

Step	Hyp	Ref	Expression
1		uncom 4160	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4347	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2786	1 ⊢ (∅ ∪ 𝐴) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 4152  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282

This theorem is used by:  sspr  4793  sstp  4794  symdifv  5049  iunxdif3  5058  nlim2  8529  indconst0  12362  pwmndid  19174  pwmnd  19175  psdmullem  22519  ltslpss  28327  leslss  28328  mulsrid  28532  mulsproplem5  28539  mulsproplem6  28540  mulsproplem7  28541  mulsproplem8  28542  coprprop  33366  fzodif1  33457  cycpmrn  33780  dflringlem3  34112  dflring4  34114  j00smo  36018  maxssj0  36019  bj-pr22val  38123  bj-snfromadj  38148  tfsconcat0i  44550  fiiuncl  46298  founiiun0  46420  infxrpnf  46671  prsal  47543  meadjun  47687  caragenuncllem  47737  carageniuncllem1  47746  hoidmvle  47825  iscnrm3rlem1  50269