Theorem difin0 3497

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 3357	. 2 |- ( A i^i B ) C_ B
2		ssdif0im 3488	. 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 1353   \ cdif 3127   i^i cin 3129   C_ wss 3130   (/)c0 3423

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424

This theorem is referenced by: (None)