Theorem dif0 3539

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 3537	. . 3 |- ( A \ A ) = (/)
2	1	difeq2i 3296	. 2 |- ( A \ ( A \ A ) ) = ( A \ (/) )
3		difdif 3306	. 2 |- ( A \ ( A \ A ) ) = A
4	2, 3	eqtr3i 2230	1 |- ( A \ (/) ) = A

Syntax hints:   = wceq 1373   \ cdif 3171   (/)c0 3468

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  disjdif2  3547  exmid1stab  4268  2oconcl  6548  diffifi  7017  undifdc  7047  difinfinf  7229  ismkvnex  7283  m1bits  12386  0cld  14699