ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssdif0im Unicode version

Theorem ssdif0im 3344
Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
ssdif0im  |-  ( A 
C_  B  ->  ( A  \  B )  =  (/) )

Proof of Theorem ssdif0im
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imanim 823 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  -.  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3006 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
31, 2sylnibr 637 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  ->  -.  x  e.  ( A  \  B ) )
43alimi 1389 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x  -.  x  e.  ( A  \  B ) )
5 dfss2 3012 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
6 eq0 3299 . 2  |-  ( ( A  \  B )  =  (/)  <->  A. x  -.  x  e.  ( A  \  B
) )
74, 5, 63imtr4i 199 1  |-  ( A 
C_  B  ->  ( A  \  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438    \ cdif 2994    C_ wss 2997   (/)c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285
This theorem is referenced by:  vdif0im  3345  difrab0eqim  3346  difid  3348  difin0  3353
  Copyright terms: Public domain W3C validator