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 2784	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1559   ∪ cun 3902  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-un 3909  df-nul 4286

This theorem is referenced by:  sspr  4792  sstp  4793  symdifv  5042  iunxdif3  5051  nlim2  8454  indconst0  12204  pwmndid  18956  pwmnd  18957  psdmullem  22210  ltslpss  27978  leslss  27979  mulsrid  28183  mulsproplem5  28190  mulsproplem6  28191  mulsproplem7  28192  mulsproplem8  28193  coprprop  32851  fzodif1  32944  cycpmrn  33284  dflringlem3  33653  dflring4  33655  bj-pr22val  37468  bj-snfromadj  37493  tfsconcat0i  43886  fiiuncl  45609  founiiun0  45732  infxrpnf  45984  prsal  46856  meadjun  47000  caragenuncllem  47050  carageniuncllem1  47059  hoidmvle  47138  iscnrm3rlem1  49525