Theorem dfdisj2 3768

Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)

Assertion

Ref	Expression
dfdisj2	|- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem dfdisj2

Step	Hyp	Ref	Expression
1		df-disj 3767	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		df-rmo 2356	. . 3 |- ( E* x e. A y e. B <-> E* x ( x e. A /\ y e. B ) )
3	2	albii 1399	. 2 |- ( A. y E* x e. A y e. B <-> A. y E* x ( x e. A /\ y e. B ) )
4	1, 3	bitri 182	1 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   e. wcel 1433   E*wmo 1942   E*wrmo 2351  Disj wdisj 3766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-rmo 2356  df-disj 3767

This theorem is referenced by:  disjss1  3772  nfdisjv  3778  invdisj  3780  sndisj  3781  disjxsn  3783