Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  undif2 Structured version   Visualization version   GIF version

Theorem undif2 3996
 Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3992). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
undif2 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)

Proof of Theorem undif2
StepHypRef Expression
1 uncom 3719 . 2 (𝐴 ∪ (𝐵𝐴)) = ((𝐵𝐴) ∪ 𝐴)
2 undif1 3995 . 2 ((𝐵𝐴) ∪ 𝐴) = (𝐵𝐴)
3 uncom 3719 . 2 (𝐵𝐴) = (𝐴𝐵)
41, 2, 33eqtri 2636 1 (𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∖ cdif 3537   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  undif  4001  dfif5  4052  funiunfv  6410  difex2  6863  undom  7933  domss2  8004  sucdom2  8041  unfi  8112  marypha1lem  8222  kmlem11  8865  hashun2  13033  hashun3  13034  cvgcmpce  14391  dprd2da  18264  dpjcntz  18274  dpjdisj  18275  dpjlsm  18276  dpjidcl  18280  ablfac1eu  18295  dfcon2  21032  2ndcdisj2  21070  fixufil  21536  fin1aufil  21546  xrge0gsumle  22444  unmbl  23112  volsup  23131  mbfss  23219  itg2cnlem2  23335  iblss2  23378  amgm  24517  wilthlem2  24595  ftalem3  24601  rpvmasum2  25001  esumpad  29444  imadifss  32554  elrfi  36275  meaunle  39357
 Copyright terms: Public domain W3C validator