Theorem dif0 3531

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 3529	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3288	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3298	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2228	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1373   \ cdif 3163   (/)c0 3460

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  disjdif2  3539  exmid1stab  4252  2oconcl  6525  diffifi  6991  undifdc  7021  difinfinf  7203  ismkvnex  7257  m1bits  12271  0cld  14584