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

Theorem ssddif 3199
Description: Double complement and subset. Similar to ddifss 3203 but inside a class  B instead of the universal class  _V. In classical logic the subset operation on the right hand side could be an equality (that is,  A  C_  B  <->  ( B  \  ( B 
\  A ) )  =  A). (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
ssddif  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )

Proof of Theorem ssddif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ancr 308 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  A
) ) )
2 simpr 107 . . . . . . . 8  |-  ( ( x  e.  B  /\  -.  x  e.  A
)  ->  -.  x  e.  A )
32con2i 567 . . . . . . 7  |-  ( x  e.  A  ->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
43anim2i 328 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 eldif 2955 . . . . . . 7  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  x  e.  ( B  \  A
) ) )
6 eldif 2955 . . . . . . . . 9  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76notbii 604 . . . . . . . 8  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 438 . . . . . . 7  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
95, 8bitri 177 . . . . . 6  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9sylibr 141 . . . . 5  |-  ( ( x  e.  B  /\  x  e.  A )  ->  x  e.  ( B 
\  ( B  \  A ) ) )
111, 10syl6 33 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
12 eldifi 3094 . . . . 5  |-  ( x  e.  ( B  \ 
( B  \  A
) )  ->  x  e.  B )
1312imim2i 12 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) )  -> 
( x  e.  A  ->  x  e.  B ) )
1411, 13impbii 121 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) ) )
1514albii 1375 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
16 dfss2 2962 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
17 dfss2 2962 . 2  |-  ( A 
C_  ( B  \ 
( B  \  A
) )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
1815, 16, 173bitr4i 205 1  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    e. wcel 1409    \ cdif 2942    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959
This theorem is referenced by:  ddifss  3203  inssddif  3206
  Copyright terms: Public domain W3C validator