Theorem dfdisj2 4622

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 4621	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		df-rmo 2920	. . 3 |- ( E* x e. A y e. B <-> E* x ( x e. A /\ y e. B ) )
3	2	albii 1747	. 2 |- ( A. y E* x e. A y e. B <-> A. y E* x ( x e. A /\ y e. B ) )
4	1, 3	bitri 264	1 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   E*wmo 2471   E*wrmo 2915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-rmo 2920  df-disj 4621

This theorem is referenced by:  disjss1  4626  nfdisj  4632  invdisj  4638  sndisj  4644  disjxsn  4646  disjss3  4652  vitalilem3  23379