Theorem disjr 3417

Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)

Assertion

Ref	Expression
disjr	|- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A )

Distinct variable groups:   x, A   x, B

Proof of Theorem disjr

Step	Hyp	Ref	Expression
1		incom 3273	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2148	. 2 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
3		disj 3416	. 2 |- ( ( B i^i A ) = (/) <-> A. x e. B -. x e. A )
4	2, 3	bitri 183	1 |- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A )

Syntax hints:   -. wn 3   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   i^i cin 3075   (/)c0 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-nul 3369

This theorem is referenced by: (None)