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Theorem dfun2 3480
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3481 and dfss4 3479 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation  \ (class difference). (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfun2  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)

Proof of Theorem dfun2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2867 . . . . . . 7  |-  x  e. 
_V
2 eldif 3238 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 884 . . . . . 6  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 676 . . . . 5  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3238 . . . . 5  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 476 . . . . 5  |-  ( -.  ( x  e.  A  \/  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 268 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  -.  (
x  e.  A  \/  x  e.  B )
)
87con2bii 322 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  -.  x  e.  ( ( _V  \  A ) 
\  B ) )
9 eldif 3238 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
101, 9mpbiran 884 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  -.  x  e.  ( ( _V  \  A )  \  B
) )
118, 10bitr4i 243 . 2  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) ) )
1211uneqri 3393 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225    u. cun 3226
This theorem is referenced by:  dfun3  3483  dfin3  3484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-dif 3231  df-un 3233
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