Theorem undif2 4428

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4423). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
undif2	⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)

Proof of Theorem undif2

Step	Hyp	Ref	Expression
1		uncom 4109	. 2 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = ((𝐵 ∖ 𝐴) ∪ 𝐴)
2		undif1 4427	. 2 ⊢ ((𝐵 ∖ 𝐴) ∪ 𝐴) = (𝐵 ∪ 𝐴)
3		uncom 4109	. 2 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
4	1, 2, 3	3eqtri 2756	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1540   ∖ cdif 3900   ∪ cun 3901

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285

This theorem is referenced by:  undif  4433  dfif5  4493  funiunfv  7184  difex2  7696  undom  8982  domss2  9053  sucdom2  9117  marypha1lem  9323  kmlem11  10055  hashun2  14290  hashun3  14291  cvgcmpce  15725  dprd2da  19923  dpjcntz  19933  dpjdisj  19934  dpjlsm  19935  dpjidcl  19939  ablfac1eu  19954  dfconn2  23304  2ndcdisj2  23342  fixufil  23807  fin1aufil  23817  xrge0gsumle  24720  unmbl  25436  volsup  25455  mbfss  25545  itg2cnlem2  25661  iblss2  25705  amgm  26899  wilthlem2  26977  ftalem3  26983  rpvmasum2  27421  noetasuplem4  27646  noetainflem4  27650  esumpad  34022  srcmpltd  35047  imadifss  37579  elrfi  42671  oaun2  43358  oaun3  43359  meaunle  46449  dfclnbgr4  47812  clnbupgr  47821