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

Theorem ssundifim 3521
Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
ssundifim  |-  ( A 
C_  ( B  u.  C )  ->  ( A  \  B )  C_  C )

Proof of Theorem ssundifim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm5.6r 928 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  \/  x  e.  C
) )  ->  (
( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
2 elun 3291 . . . . 5  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
32imbi2i 226 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  ->  ( x  e.  B  \/  x  e.  C ) ) )
4 eldif 3153 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54imbi1i 238 . . . 4  |-  ( ( x  e.  ( A 
\  B )  ->  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  C ) )
61, 3, 53imtr4i 201 . . 3  |-  ( ( x  e.  A  ->  x  e.  ( B  u.  C ) )  -> 
( x  e.  ( A  \  B )  ->  x  e.  C
) )
76alimi 1466 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  ( B  u.  C
) )  ->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
8 dfss2 3159 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A. x
( x  e.  A  ->  x  e.  ( B  u.  C ) ) )
9 dfss2 3159 . 2  |-  ( ( A  \  B ) 
C_  C  <->  A. x
( x  e.  ( A  \  B )  ->  x  e.  C
) )
107, 8, 93imtr4i 201 1  |-  ( A 
C_  ( B  u.  C )  ->  ( A  \  B )  C_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709   A.wal 1362    e. wcel 2160    \ cdif 3141    u. cun 3142    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-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator