Theorem 0un 4419

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 4181	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4417	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2768	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1537   ∪ cun 3974  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353

This theorem is referenced by:  nlim2  8546  pwmndid  18971  pwmnd  18972  psdmullem  22192  sltlpss  27963  slelss  27964  mulsrid  28157  mulsproplem5  28164  mulsproplem6  28165  mulsproplem7  28166  mulsproplem8  28167  coprprop  32711  fzodif1  32798  cycpmrn  33136  bj-pr22val  36985  bj-snfromadj  37010  metakunt17  42178  tfsconcat0i  43307  fiiuncl  44967  founiiun0  45097  infxrpnf  45361  prsal  46239  meadjun  46383  caragenuncllem  46433  carageniuncllem1  46442  hoidmvle  46521  iscnrm3rlem1  48620