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Theorem dif0 3403
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 3401 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3161 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3171 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2140 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    \ cdif 3038   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  disjdif2  3411  2oconcl  6304  diffifi  6756  undifdc  6780  difinfinf  6954  ismkvnex  6997  0cld  12208  exmid1stab  13122
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