For that use the dplyr package and with the help of the package, just group it by the columns that the dataset is having:
library(dplyr)
df %>% group_by(Ori., Dest) %>% filter(n() == 1)
# Ori. Dest Carr. Pass Flights
# <chr> <chr> <chr> <int> <int>
#1 JFK BOS Delta 15344 89
#2 ATL FLR AmerA 25054 90
subset(df, ave(Flights, Ori., Dest, FUN = length) == 1)
data
df <- structure(list(Ori. = c("JFK", "JFK", "JFK", "ATL", "OHD", "OHD"
), Dest = c("LAX", "LAX", "BOS", "FLR", "LAX", "LAX"), Carr. = c("Delta",
"JetBl", "Delta", "AmerA", "Delta", "AmerA"), Pass = c(15004L,
17434L, 15344L, 25054L, 19876L, 12344L), Flights = c(50L, 100L,
89L, 90L, 95L, 45L)), class = "data.frame", row.names = c(NA, -6L))