Theorem dif0 3594

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 3592	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3344	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3354	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2261	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1402   \ cdif 3217   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  disjdif2  3603  exmid1stab  4340  2oconcl  6702  diffifi  7188  undifdc  7221  difinfinf  7431  ismkvnex  7485  m1bits  12705  0cld  15136