ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difin2 Unicode version

Theorem difin2 3412
Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )

Proof of Theorem difin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3164 . . . . 5  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
21pm4.71d 393 . . . 4  |-  ( A 
C_  C  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  C ) ) )
32anbi1d 465 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B )
) )
4 eldif 3153 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3333 . . . 4  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( x  e.  ( C  \  B
)  /\  x  e.  A ) )
6 eldif 3153 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
76anbi1i 458 . . . 4  |-  ( ( x  e.  ( C 
\  B )  /\  x  e.  A )  <->  ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A ) )
8 ancom 266 . . . . 5  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
9 anass 401 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
108, 9bitr4i 187 . . . 4  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
115, 7, 103bitri 206 . . 3  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
123, 4, 113bitr4g 223 . 2  |-  ( A 
C_  C  ->  (
x  e.  ( A 
\  B )  <->  x  e.  ( ( C  \  B )  i^i  A
) ) )
1312eqrdv 2187 1  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    \ cdif 3141    i^i cin 3143    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator