Theorem disjr 3518

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 3373	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2215	. 2 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
3		disj 3517	. 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 1373   e. wcel 2178   A.wral 2486   i^i cin 3173   (/)c0 3468

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-nul 3469

This theorem is referenced by: (None)