Theorem dfdisj2 4108

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 4107	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		df-rmo 2536	. . 3 |- ( E* x e. A y e. B <-> E* x ( x e. A /\ y e. B ) )
3	2	albii 1523	. 2 |- ( A. y E* x e. A y e. B <-> A. y E* x ( x e. A /\ y e. B ) )
4	1, 3	bitri 184	1 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   A.wal 1400   E*wmo 2087   e. wcel 2209   E*wrmo 2531  Disj wdisj 4106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502

This proof depends on definitions:  df-bi 117  df-rmo 2536  df-disj 4107

This theorem is used by:  disjss1  4112  nfdisjv  4118  invdisj  4123  sndisj  4126  disjxsn  4128