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

Step	Hyp	Ref	Expression
1		difid 3357	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3118	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3128	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2111	1 |- ( A \ (/) ) = A

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