Theorem disjr 3486

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 3341	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2196	. 2 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
3		disj 3485	. 2 |- ( ( B i^i A ) = (/) <-> A. x e. B -. x e. A )
4	2, 3	bitri 184	1 |- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A )

Syntax hints:   -. wn 3   <-> wb 105   = wceq 1363   e. wcel 2159   A.wral 2467   i^i cin 3142   (/)c0 3436

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-v 2753  df-dif 3145  df-in 3149  df-nul 3437

This theorem is referenced by: (None)