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Theorem undif1 3833
Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3830). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
undif1  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)

Proof of Theorem undif1
StepHypRef Expression
1 undir 3683 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  B )  =  ( ( A  u.  B )  i^i  (
( _V  \  B
)  u.  B ) )
2 invdif 3675 . . 3  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
32uneq1i 3575 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  B )  =  ( ( A  \  B )  u.  B
)
4 uncom 3569 . . . . 5  |-  ( ( _V  \  B )  u.  B )  =  ( B  u.  ( _V  \  B ) )
5 unvdif 3832 . . . . 5  |-  ( B  u.  ( _V  \  B ) )  =  _V
64, 5eqtri 2493 . . . 4  |-  ( ( _V  \  B )  u.  B )  =  _V
76ineq2i 3622 . . 3  |-  ( ( A  u.  B )  i^i  ( ( _V 
\  B )  u.  B ) )  =  ( ( A  u.  B )  i^i  _V )
8 inv1 3764 . . 3  |-  ( ( A  u.  B )  i^i  _V )  =  ( A  u.  B
)
97, 8eqtri 2493 . 2  |-  ( ( A  u.  B )  i^i  ( ( _V 
\  B )  u.  B ) )  =  ( A  u.  B
)
101, 3, 93eqtr3i 2501 1  |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723
This theorem is referenced by:  undif2  3834  unidif0  4574  pwundif  4746  sofld  5290  fresaun  5766  ralxpmap  7539  enp1ilem  7823  difinf  7859  pwfilem  7886  infdif  8657  fin23lem11  8765  fin1a2lem13  8860  axcclem  8905  ttukeylem1  8957  ttukeylem7  8963  fpwwe2lem13  9085  hashbclem  12656  incexclem  13971  ramub1lem1  15063  ramub1lem2  15064  isstruct2  15208  mrieqvlemd  15613  mreexmrid  15627  islbs3  18456  lbsextlem4  18462  basdif0  20045  bwth  20502  locfincmp  20618  cldsubg  21203  nulmbl2  22568  volinun  22578  limcdif  22910  ellimc2  22911  limcmpt2  22918  dvreslem  22943  dvaddbr  22971  dvmulbr  22972  lhop  23047  plyeq0  23244  rlimcnp  23970  difeq  28230  ffsrn  28389  esumpad2  28951  measunl  29112  subfacp1lem1  29974  cvmscld  30068  compne  36863  stoweidlem44  38017
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