Theorem dif0 3314

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

Step	Hyp	Ref	Expression
1		difid 3312	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3087	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3097	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2103	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1284   \ cdif 2970   (/)c0 3251

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by:  2oconcl  6045  diffifi  6378