Metamath Proof Explorer


Theorem dissnref

Description: The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020)

Ref Expression
Hypothesis dissnref.c C=u|xXu=x
Assertion dissnref XVY=XCRefY

Proof

Step Hyp Ref Expression
1 dissnref.c C=u|xXu=x
2 simpr XVY=XY=X
3 1 unisngl X=C
4 2 3 eqtrdi XVY=XY=C
5 simplr XVY=XuCxXu=xyYxyu=x
6 simprr XVY=XuCxXu=xyYxyxy
7 6 snssd XVY=XuCxXu=xyYxyxy
8 5 7 eqsstrd XVY=XuCxXu=xyYxyuy
9 simplr XVY=XuCxXu=xxX
10 simp-4r XVY=XuCxXu=xY=X
11 9 10 eleqtrrd XVY=XuCxXu=xxY
12 eluni2 xYyYxy
13 11 12 sylib XVY=XuCxXu=xyYxy
14 8 13 reximddv XVY=XuCxXu=xyYuy
15 1 eqabri uCxXu=x
16 15 biimpi uCxXu=x
17 16 adantl XVY=XuCxXu=x
18 14 17 r19.29a XVY=XuCyYuy
19 18 ralrimiva XVY=XuCyYuy
20 pwexg XV𝒫XV
21 simpr uCxXu=xu=x
22 snelpwi xXx𝒫X
23 22 ad2antlr uCxXu=xx𝒫X
24 21 23 eqeltrd uCxXu=xu𝒫X
25 24 16 r19.29a uCu𝒫X
26 25 ssriv C𝒫X
27 26 a1i XVC𝒫X
28 20 27 ssexd XVCV
29 28 adantr XVY=XCV
30 eqid C=C
31 eqid Y=Y
32 30 31 isref CVCRefYY=CuCyYuy
33 29 32 syl XVY=XCRefYY=CuCyYuy
34 4 19 33 mpbir2and XVY=XCRefY