Theorem disjr 3464

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 3319	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2178	. 2 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
3		disj 3463	. 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 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415

This theorem is referenced by: (None)