Theorem dif0 3485

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 3483	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3242	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3252	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2193	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1348   \ cdif 3118   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  disjdif2  3493  2oconcl  6418  diffifi  6872  undifdc  6901  difinfinf  7078  ismkvnex  7131  0cld  12906  exmid1stab  14033