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

Theorem symdifxor 3263
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 3006 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3006 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 716 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3139 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 excxor 1314 . . . 4  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
) )
6 ancom 262 . . . . 5  |-  ( ( -.  x  e.  A  /\  x  e.  B
)  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76orbi2i 714 . . . 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 182 . . 3  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
93, 4, 83bitr4i 210 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  A  \/_  x  e.  B ) )
109abbi2i 2202 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    \/ wo 664    = wceq 1289    \/_ wxo 1311    e. wcel 1438   {cab 2074    \ cdif 2994    u. cun 2995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-xor 1312  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator