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Metamath Proof Explorer

Theorem disjenex

Description: Existence version of disjen . (Contributed by Mario Carneiro, 7-Feb-2015)

Ref Expression
Assertion disjenex 
|- ( ( A e. V /\ B e. W ) -> E. x ( ( A i^i x ) = (/) /\ x ~~ B ) )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( A e. V /\ B e. W ) -> B e. W )
2 snex 
 |-  { ~P U. ran A } e. _V
3 xpexg 
 |-  ( ( B e. W /\ { ~P U. ran A } e. _V ) -> ( B X. { ~P U. ran A } ) e. _V )
4 1 2 3 sylancl 
 |-  ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) e. _V )
5 disjen 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) )
6 ineq2 
 |-  ( x = ( B X. { ~P U. ran A } ) -> ( A i^i x ) = ( A i^i ( B X. { ~P U. ran A } ) ) )
7 6 eqeq1d 
 |-  ( x = ( B X. { ~P U. ran A } ) -> ( ( A i^i x ) = (/) <-> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) ) )
8 breq1 
 |-  ( x = ( B X. { ~P U. ran A } ) -> ( x ~~ B <-> ( B X. { ~P U. ran A } ) ~~ B ) )
9 7 8 anbi12d 
 |-  ( x = ( B X. { ~P U. ran A } ) -> ( ( ( A i^i x ) = (/) /\ x ~~ B ) <-> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) ) )
10 4 5 9 spcedv 
 |-  ( ( A e. V /\ B e. W ) -> E. x ( ( A i^i x ) = (/) /\ x ~~ B ) )