Metamath Proof Explorer


Theorem fneint

Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009)

Ref Expression
Assertion fneint A Fne B x B | P x x A | P x

Proof

Step Hyp Ref Expression
1 eleq2w x = y P x P y
2 1 elrab y x A | P x y A P y
3 fnessex A Fne B y A P y z B P z z y
4 3 3expb A Fne B y A P y z B P z z y
5 eleq2w x = z P x P z
6 5 intminss z B P z x B | P x z
7 sstr x B | P x z z y x B | P x y
8 6 7 sylan z B P z z y x B | P x y
9 8 expl z B P z z y x B | P x y
10 9 rexlimiv z B P z z y x B | P x y
11 4 10 syl A Fne B y A P y x B | P x y
12 11 ex A Fne B y A P y x B | P x y
13 2 12 syl5bi A Fne B y x A | P x x B | P x y
14 13 ralrimiv A Fne B y x A | P x x B | P x y
15 ssint x B | P x x A | P x y x A | P x x B | P x y
16 14 15 sylibr A Fne B x B | P x x A | P x