Theorem dfdisj2 3968

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 3967	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		df-rmo 2456	. . 3 |- ( E* x e. A y e. B <-> E* x ( x e. A /\ y e. B ) )
3	2	albii 1463	. 2 |- ( A. y E* x e. A y e. B <-> A. y E* x ( x e. A /\ y e. B ) )
4	1, 3	bitri 183	1 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1346   E*wmo 2020   e. wcel 2141   E*wrmo 2451  Disj wdisj 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442

This theorem depends on definitions:  df-bi 116  df-rmo 2456  df-disj 3967

This theorem is referenced by:  disjss1  3972  nfdisjv  3978  invdisj  3983  sndisj  3985  disjxsn  3987