Metamath Proof Explorer


Theorem elblps

Description: Membership in a ball. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013) (Revised by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion elblps D PsMet X P X R * A P ball D R A X P D A < R

Proof

Step Hyp Ref Expression
1 blvalps D PsMet X P X R * P ball D R = x X | P D x < R
2 1 eleq2d D PsMet X P X R * A P ball D R A x X | P D x < R
3 oveq2 x = A P D x = P D A
4 3 breq1d x = A P D x < R P D A < R
5 4 elrab A x X | P D x < R A X P D A < R
6 2 5 bitrdi D PsMet X P X R * A P ball D R A X P D A < R