Theorem 0un 4371

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 4133	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4369	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2758	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1540   ∪ cun 3924  ∅c0 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-nul 4309

This theorem is referenced by:  nlim2  8502  pwmndid  18914  pwmnd  18915  psdmullem  22103  sltlpss  27871  slelss  27872  mulsrid  28068  mulsproplem5  28075  mulsproplem6  28076  mulsproplem7  28077  mulsproplem8  28078  coprprop  32676  fzodif1  32769  cycpmrn  33154  bj-pr22val  37037  bj-snfromadj  37062  metakunt17  42234  tfsconcat0i  43369  fiiuncl  45089  founiiun0  45214  infxrpnf  45473  prsal  46347  meadjun  46491  caragenuncllem  46541  carageniuncllem1  46550  hoidmvle  46629  iscnrm3rlem1  48914