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Theorem dif0 3508
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 3506 . . 3  |-  ( A 
\  A )  =  (/)
21difeq2i 3265 . 2  |-  ( A 
\  ( A  \  A ) )  =  ( A  \  (/) )
3 difdif 3275 . 2  |-  ( A 
\  ( A  \  A ) )  =  A
42, 3eqtr3i 2212 1  |-  ( A 
\  (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1364    \ cdif 3141   (/)c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438
This theorem is referenced by:  disjdif2  3516  exmid1stab  4226  2oconcl  6463  diffifi  6921  undifdc  6951  difinfinf  7129  ismkvnex  7182  0cld  14064
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