Theorem difin0 3542

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 3402	. 2 |- ( A i^i B ) C_ B
2		ssdif0im 3533	. 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 1373   \ cdif 3171   i^i cin 3173   C_ wss 3174   (/)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-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by: (None)