ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  symdifxor Unicode version

Theorem symdifxor 3266
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
symdifxor  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem symdifxor
StepHypRef Expression
1 eldif 3009 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3009 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 717 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3142 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 excxor 1315 . . . 4  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
) )
6 ancom 263 . . . . 5  |-  ( ( -.  x  e.  A  /\  x  e.  B
)  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76orbi2i 715 . . . 4  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
)  <->  ( ( x  e.  A  /\  -.  x  e.  B )  \/  ( x  e.  B  /\  -.  x  e.  A
) ) )
85, 7bitri 183 . . 3  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
93, 4, 83bitr4i 211 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  A  \/_  x  e.  B ) )
109abbi2i 2203 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 665    = wceq 1290    \/_ wxo 1312    e. wcel 1439   {cab 2075    \ cdif 2997    u. cun 2998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-xor 1313  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-un 3004
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator