Theorem disj2 3824

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 3464	. 2 |- A C_ _V
2		reldisj 3820	. 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 189   = wceq 1455   _Vcvv 3057   \ cdif 3413   i^i cin 3415   C_ wss 3416   (/)c0 3743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744

This theorem is referenced by:  ssindif0  3830  intirr  5240  setsres  15206  setscom  15208  f1omvdco3  17145  psgnunilem5  17190  opsrtoslem2  18763  clscon  20500  cldsubg  21180  uniinn0  28218  imadifxp  28265