Theorem disj2 3874

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 3524	. 2 |- A C_ _V
2		reldisj 3870	. 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 184   = wceq 1379   _Vcvv 3113   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786

This theorem is referenced by:  ssindif0  3880  intirr  5383  setsres  14511  setscom  14513  f1omvdco3  16267  psgnunilem5  16312  opsrtoslem2  17917  clscon  19694  cldsubg  20341  imadifxp  27128