Theorem disj2 3342

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 3047	. 2 |- A C_ _V
2		reldisj 3338	. 2 |- ( A C_ _V -> ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) ) )
3	1, 2	ax-mp 7	1 |- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) )

Syntax hints:   <-> wb 104   = wceq 1290   _Vcvv 2620   \ cdif 2997   i^i cin 2999   C_ wss 3000   (/)c0 3287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288

This theorem is referenced by:  ssindif0im  3346  intirr  4831  setsresg  11593  setscom  11595