You can also do this with the pipe operator, %>%, from the magrittr package.
library(magrittr)
DF <- data.frame(Name = LETTERS[1:5], Value = c(1,2,NA,4,5))
DF
#> Name Value
#> 1 A 1
#> 2 B 2
#> 3 C NA
#> 4 D 4
#> 5 E 5
DF_New <- DF %>% na.omit() %>% subset(Value > 1 & Value < 5)
DF_New
#> Name Value
#> 2 B 2
#> 4 D 4
Created on 2019-09-29 by the reprex package (v0.2.1)