Theorem disjr 3573

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 3421	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2246	. 2 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
3		disj 3572	. 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 1402   e. wcel 2209   A.wral 2528   i^i cin 3219   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-in 3226  df-nul 3521

This theorem is referenced by: (None)