Theorem difun2 4048

Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
difun2	|- ( ( A u. B ) \ B ) = ( A \ B )

Proof of Theorem difun2

Step	Hyp	Ref	Expression
1		difundir 3880	. 2 |- ( ( A u. B ) \ B ) = ( ( A \ B ) u. ( B \ B ) )
2		difid 3948	. . 3 |- ( B \ B ) = (/)
3	2	uneq2i 3764	. 2 |- ( ( A \ B ) u. ( B \ B ) ) = ( ( A \ B ) u. (/) )
4		un0 3967	. 2 |- ( ( A \ B ) u. (/) ) = ( A \ B )
5	1, 3, 4	3eqtri 2648	1 |- ( ( A u. B ) \ B ) = ( A \ B )

Syntax hints:   = wceq 1483   \ cdif 3571   u. cun 3572   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  uneqdifeq  4057  uneqdifeqOLD  4058  difprsn1  4330  orddif  5820  domunsncan  8060  elfiun  8336  hartogslem1  8447  cantnfp1lem3  8577  cda1dif  8998  infcda1  9015  ssxr  10107  dfn2  11305  incexclem  14568  mreexmrid  16303  lbsextlem4  19161  ufprim  21713  volun  23313  i1f1  23457  itgioo  23582  itgsplitioo  23604  plyeq0lem  23966  jensen  24715  difeq  29355  fzdif2  29551  fzodif2  29552  measun  30274  carsgclctunlem1  30379  carsggect  30380  chtvalz  30707  elmrsubrn  31417  mrsubvrs  31419  finixpnum  33394  lindsenlbs  33404  poimirlem2  33411  poimirlem4  33413  poimirlem6  33415  poimirlem7  33416  poimirlem8  33417  poimirlem11  33420  poimirlem12  33421  poimirlem13  33422  poimirlem14  33423  poimirlem16  33425  poimirlem18  33427  poimirlem19  33428  poimirlem21  33430  poimirlem23  33432  poimirlem27  33436  poimirlem30  33439  asindmre  33495  kelac2  37635  pwfi2f1o  37666  iccdifioo  39741  iccdifprioo  39742  hoiprodp1  40802