While the F' plasmid does contain the ccdA gene that can inhibit or reduce the toxicity of the ccdB gene product, the ccdA expression level is likely to be too low, or inhibition may not be complete, and the bacteria would still be exposed to the ccdB gene product and thus not grow. Therefore, bacterial strains containing the F' plasmid are not recommended as hosts for propagation of ccdB containing vectors.
For propagation of Gateway vectors containing ccdB, we recommend the One Shot ccdB Survival 2 T1R Competent Cells (A10460), which were specifically designed for that purpose. However, please note that these cells are not validated for propagation of other ccdB-containing vectors like the older pZErO plasmids, and in most cases they are not expected to work due to very high levels of ccdB protein expressed in those vectors.