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Theorem dfun2 3738
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3739 and dfss4 3737 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 3121 . . . . . . 7  |-  x  e. 
_V
2 eldif 3491 . . . . . . 7  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 916 . . . . . 6  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43anbi1i 695 . . . . 5  |-  ( ( x  e.  ( _V 
\  A )  /\  -.  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
5 eldif 3491 . . . . 5  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  ( x  e.  ( _V  \  A
)  /\  -.  x  e.  B ) )
6 ioran 490 . . . . 5  |-  ( -.  ( x  e.  A  \/  x  e.  B
)  <->  ( -.  x  e.  A  /\  -.  x  e.  B ) )
74, 5, 63bitr4i 277 . . . 4  |-  ( x  e.  ( ( _V 
\  A )  \  B )  <->  -.  (
x  e.  A  \/  x  e.  B )
)
87con2bii 332 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  -.  x  e.  ( ( _V  \  A ) 
\  B ) )
9 eldif 3491 . . . 4  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( ( _V  \  A )  \  B
) ) )
101, 9mpbiran 916 . . 3  |-  ( x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) )  <->  -.  x  e.  ( ( _V  \  A )  \  B
) )
118, 10bitr4i 252 . 2  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  ( _V  \ 
( ( _V  \  A )  \  B
) ) )
1211uneqri 3651 1  |-  ( A  u.  B )  =  ( _V  \  (
( _V  \  A
)  \  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    u. cun 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-dif 3484  df-un 3486
This theorem is referenced by:  dfun3  3741  dfin3  3742
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