Theorem dif0 3495

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 3493	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3252	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3262	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2200	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1353   \ cdif 3128   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425

This theorem is referenced by:  disjdif2  3503  exmid1stab  4210  2oconcl  6442  diffifi  6896  undifdc  6925  difinfinf  7102  ismkvnex  7155  0cld  13697