Theorem difin0 3482

Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difin0	|- ( ( A i^i B ) \ B ) = (/)

Proof of Theorem difin0

Step	Hyp	Ref	Expression
1		inss2 3343	. 2 |- ( A i^i B ) C_ B
2		ssdif0im 3473	. 2 |- ( ( A i^i B ) C_ B -> ( ( A i^i B ) \ B ) = (/) )
3	1, 2	ax-mp 5	1 |- ( ( A i^i B ) \ B ) = (/)

Syntax hints:   = wceq 1343   \ cdif 3113   i^i cin 3115   C_ wss 3116   (/)c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by: (None)