R How to use apply with ifelse function to search for string values over many columns?

I know there are many similar questions to this but just can't figure this out.

I want an ifelse function to go over many columns in a dataframe. I want to add two variables to the dataframe, "C03_only" and "only_c02_and_c09". I am only focused on entries that contain values: "C02 ","C03", "C09".

Example data:

mydf<- data.frame(id=1:4,
                  x1=c("A02", "C02", "C03", "M01"),
                  x2=c("B02", "", "C02", "C09"),
                  x3=c("C03", "C03", "C09", "C02") )

R>mydf
  id  x1  x2  x3
1  1 A02 B02 C03
2  2 C02     C03
3  3 C03 C02 C09
4  4 M01 C09 C02

The new dataset should look like:

R>mydf
  id  x1  x2  x3 C03_only only_c02_and_c09
1  1 A02 B02 C03        1                0
2  2 C02     C03        0                0
3  3 C03 C02 C09        0                0
4  4 M01 C09 C02        0                1

I first tried something like this

mydf$C03_only <- with(mydf,ifelse(x1 != "C02" | "C09" & x2 !="C02" | "C09" & x3== "C03",1,0))

which didnt work but the idea is terrible as I have many columns so is a no runner. Similarly I tired something with a for loop:

mydf$C03_only<-rep(0,nrow(mydf))
for (i in 2:nrow(mydf)){
  if (mydf$x1[i]!="C02" && mydf$x2[i]!="C09" && mydf$x3[i]=="C03"){
    mydf$C03_only[i]<-1}
}

This also didnt work but (only partially finished) with enough playing with it, it probably would.

I think the best approach is to use apply function but can't get it working:

mydf$C03_only<- apply(mydf[,-1], MARGIN=1, FUN=function(x){ 
  ifelse(any(x == "C03") & any(x != "C09" & x != "C02") , 1, 0)
}
)

mydf$only_c02_and_c09<- apply(mydf[,-1], MARGIN=1, FUN=function(x){ 
  ifelse(any(x == "C02" & x == "C09") & any(x != "C03") , 1, 0)
}
)

These are close but no cigar. I need to replace any with something but not sure what. Perhaps pass the variables of interest to a vector and run some conditional statement using %in% on this but I'm not sure how.

Any suggestions would be great, thanks.