Theorem 0un 4332

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 4092	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4330	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2768	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1542   ∪ cun 3890  ∅c0 4262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263

This theorem is referenced by:  pwmndid  18573  pwmnd  18574  coprprop  31028  fzodif1  31110  cycpmrn  31406  sltlpss  34083  bj-pr22val  35205  metakunt17  40138  fiiuncl  42583  founiiun0  42698  infxrpnf  42957  prsal  43830  meadjun  43971  caragenuncllem  44021  carageniuncllem1  44030  hoidmvle  44109  iscnrm3rlem1  46203