Theorem disjr 15675

Description: Two ways of saying that two classes are disjoint.

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 2787	. . 3 |- (A i^i B) = (B i^i A)
2	1	eqeq1i 1891	. 2 |- ((A i^i B) = (/) <-> (B i^i A) = (/))
3		disj 2914	. 2 |- ((B i^i A) = (/) <-> A.x e. B -. x e. A)
4	2, 3	bitri 190	1 |- ((A i^i B) = (/) <-> A.x e. B -. x e. A)

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592  (/)c0 2875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-dif 2597  df-in 2603  df-nul 2876

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