Theorem dif0 3562

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 3560	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3319	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3329	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2252	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1395   \ cdif 3194   (/)c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by:  disjdif2  3570  exmid1stab  4292  2oconcl  6585  diffifi  7056  undifdc  7086  difinfinf  7268  ismkvnex  7322  m1bits  12471  0cld  14786