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

Theorem dif0 3433
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 3431 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3191 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3201 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2162 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1331    \ cdif 3068   (/)c0 3363
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364
This theorem is referenced by:  disjdif2  3441  2oconcl  6336  diffifi  6788  undifdc  6812  difinfinf  6986  ismkvnex  7029  0cld  12291  exmid1stab  13225
  Copyright terms: Public domain W3C validator