Theorem disj2 3413

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 3114	. 2 |- A C_ _V
2		reldisj 3409	. 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 1331   _Vcvv 2681   \ cdif 3063   i^i cin 3065   C_ wss 3066   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by:  ssindif0im  3417  intirr  4920  setsresg  11986  setscom  11988