Theorem dfdisj2 3747

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 3746	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		df-rmo 2314	. . 3 |- ( E* x e. A y e. B <-> E* x ( x e. A /\ y e. B ) )
3	2	albii 1359	. 2 |- ( A. y E* x e. A y e. B <-> A. y E* x ( x e. A /\ y e. B ) )
4	1, 3	bitri 173	1 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   e. wcel 1393   E*wmo 1901   E*wrmo 2309  Disj wdisj 3745

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-rmo 2314  df-disj 3746

This theorem is referenced by:  disjss1  3751  nfdisjv  3757  invdisj  3759  sndisj  3760  disjxsn  3762