Theorem disj2 3842

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 3484	. 2 |- A C_ _V
2		reldisj 3838	. 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 187   = wceq 1437   _Vcvv 3080   \ cdif 3433   i^i cin 3435   C_ wss 3436   (/)c0 3761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762

This theorem is referenced by:  ssindif0  3848  intirr  5237  setsres  15150  setscom  15152  f1omvdco3  17089  psgnunilem5  17134  opsrtoslem2  18707  clscon  20443  cldsubg  21123  uniinn0  28165  imadifxp  28214