Theorem dif0 3517

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 3515	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3274	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3284	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2216	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1364   \ cdif 3150   (/)c0 3446

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by:  disjdif2  3525  exmid1stab  4237  2oconcl  6492  diffifi  6950  undifdc  6980  difinfinf  7160  ismkvnex  7214  0cld  14280