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

Theorem difdifdirss 3415
Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
difdifdirss  |-  ( ( A  \  B ) 
\  C )  C_  ( ( A  \  C )  \  ( B  \  C ) )

Proof of Theorem difdifdirss
StepHypRef Expression
1 dif32 3307 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3286 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2139 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3364 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2139 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3291 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3403 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3236 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2138 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3195 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2139 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2139 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddifss 3282 . . . . . 6  |-  C  C_  ( _V  \  ( _V  \  C ) )
14 unss2 3215 . . . . . 6  |-  ( C 
C_  ( _V  \ 
( _V  \  C
) )  ->  (
( _V  \  B
)  u.  C ) 
C_  ( ( _V 
\  B )  u.  ( _V  \  ( _V  \  C ) ) ) )
1513, 14ax-mp 5 . . . . 5  |-  ( ( _V  \  B )  u.  C )  C_  ( ( _V  \  B )  u.  ( _V  \  ( _V  \  C ) ) )
16 indmss 3303 . . . . . 6  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  i^i  ( _V  \  C ) ) )
17 invdif 3286 . . . . . . 7  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1817difeq2i 3159 . . . . . 6  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1916, 18sseqtri 3099 . . . . 5  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  C_  ( _V  \  ( B  \  C ) )
2015, 19sstri 3074 . . . 4  |-  ( ( _V  \  B )  u.  C )  C_  ( _V  \  ( B  \  C ) )
21 sslin 3270 . . . 4  |-  ( ( ( _V  \  B
)  u.  C ) 
C_  ( _V  \ 
( B  \  C
) )  ->  (
( A  \  C
)  i^i  ( ( _V  \  B )  u.  C ) )  C_  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) ) )
2220, 21ax-mp 5 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  C_  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) )
23 invdif 3286 . . 3  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2422, 23sseqtri 3099 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  C_  ( ( A  \  C )  \  ( B  \  C ) )
2512, 24eqsstri 3097 1  |-  ( ( A  \  B ) 
\  C )  C_  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2658    \ cdif 3036    u. cun 3037    i^i cin 3038    C_ wss 3039   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator