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Theorem undif2 3491
Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3487). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
undif2  |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B
)

Proof of Theorem undif2
StepHypRef Expression
1 uncom 3280 . 2  |-  ( A  u.  ( B  \  A ) )  =  ( ( B  \  A )  u.  A
)
2 undif1 3490 . 2  |-  ( ( B  \  A )  u.  A )  =  ( B  u.  A
)
3 uncom 3280 . 2  |-  ( B  u.  A )  =  ( A  u.  B
)
41, 2, 33eqtri 2280 1  |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1619    \ cdif 3110    u. cun 3111
This theorem is referenced by:  undif  3495  dfif5  3537  difex2  4483  funiunfv  5694  undom  6904  domss2  6974  sucdom2  7011  unfi  7078  marypha1lem  7140  kmlem11  7740  hashun2  11317  cvgcmpce  12227  dprd2da  15225  dpjcntz  15235  dpjdisj  15236  dpjlsm  15237  dpjidcl  15241  ablfac1eu  15256  dfcon2  17093  2ndcdisj2  17131  fixufil  17565  fin1aufil  17575  xrge0gsumle  18286  unmbl  18843  volsup  18861  mbfss  18949  itg2cnlem2  19065  iblss2  19108  amgm  20233  wilthlem2  20255  ftalem3  20260  rpvmasum2  20609  elrfi  26122
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417
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