Theorem disj2 3520

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 3219	. 2 |- A C_ _V
2		reldisj 3516	. 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 105   = wceq 1373   _Vcvv 2773   \ cdif 3167   i^i cin 3169   C_ wss 3170   (/)c0 3464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-nul 3465

This theorem is referenced by:  ssindif0im  3524  intirr  5078  setsresg  12945  setscom  12947