Omit rows containing specific column of NA

I want to know how to omit NA values in a data frame, but only in some columns, I am interested in.

For example,

DF <- data.frame(x = c(1, 2, 3), y = c(0, 10, NA), z=c(NA, 33, 22))

but I only want to omit the data where y is NA, therefore the result should be

  x  y  z
1 1  0 NA
2 2 10 33

na.omit seems delete all rows contain any NA.

Can somebody help me out of this simple question?

But if now I change the question like:

DF <- data.frame(x = c(1, 2, 3,NA), y = c(1,0, 10, NA), z=c(43,NA, 33, NA))

If I want to omit only x=na or z=na, where can I put the | in function?