Theorem disj2 3423

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
disj2	|- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) )

Proof of Theorem disj2

Step	Hyp	Ref	Expression
1		ssv 3124	. 2 |- A C_ _V
2		reldisj 3419	. 2 |- ( A C_ _V -> ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) ) )
3	1, 2	ax-mp 5	1 |- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) )

Syntax hints:   <-> wb 104   = wceq 1332   _Vcvv 2689   \ cdif 3073   i^i cin 3075   C_ wss 3076   (/)c0 3368

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by:  ssindif0im  3427  intirr  4933  setsresg  12036  setscom  12038