ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dif0 Unicode version

Theorem dif0 3359
Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
dif0  |-  ( A 
\  (/) )  =  A

Proof of Theorem dif0
StepHypRef Expression
1 difid 3357 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3118 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3128 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2111 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1290    \ cdif 2999   (/)c0 3289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rab 2369  df-v 2624  df-dif 3004  df-in 3008  df-ss 3015  df-nul 3290
This theorem is referenced by:  disjdif2  3367  2oconcl  6219  diffifi  6666  undifdc  6690  0cld  11875
  Copyright terms: Public domain W3C validator