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 2761	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1542   ∪ cun 3947  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324

This theorem is referenced by:  nlim2  8490  pwmndid  18817  pwmnd  18818  sltlpss  27401  mulsrid  27569  mulsproplem5  27576  mulsproplem6  27577  mulsproplem7  27578  mulsproplem8  27579  coprprop  31921  fzodif1  32004  cycpmrn  32302  bj-pr22val  35900  bj-snfromadj  35925  metakunt17  41001  tfsconcat0i  42095  fiiuncl  43752  founiiun0  43888  infxrpnf  44156  prsal  45034  meadjun  45178  caragenuncllem  45228  carageniuncllem1  45237  hoidmvle  45316  iscnrm3rlem1  47573