Theorem 0un 4270

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 4054	. 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
2		un0 4268	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
3	1, 2	eqtri 2819	1 ⊢ (∅ ∪ 𝐴) = 𝐴

Syntax hints:   = wceq 1522   ∪ cun 3861  ∅c0 4215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3866  df-un 3868  df-nul 4216

This theorem is referenced by:  coprprop  30128  fzodif1  30207  cycpmrn  30427  fiiuncl  40892  founiiun0  41017  infxrpnf  41289  prsal  42172  meadjun  42313  caragenuncllem  42363  carageniuncllem1  42372  hoidmvle  42451