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

Step	Hyp	Ref	Expression
1		difid 3506	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3265	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3275	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2212	1 |- ( A \ (/) ) = A

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