MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdif2 Unicode version

Theorem dfdif2 3161
Description: Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfdif2  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
Distinct variable groups:    x, A    x, B

Proof of Theorem dfdif2
StepHypRef Expression
1 df-dif 3155 . 2  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
2 df-rab 2552 . 2  |-  { x  e.  A  |  -.  x  e.  B }  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
31, 2eqtr4i 2306 1  |-  ( A 
\  B )  =  { x  e.  A  |  -.  x  e.  B }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    \ cdif 3149
This theorem is referenced by:  difeq1  3287  difeq2  3288  nfdif  3297  difidALT  3523  ordintdif  4441  kmlem3  7778  incexc2  12297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-cleq 2276  df-rab 2552  df-dif 3155
  Copyright terms: Public domain W3C validator