Rav Shlomo Zalman Auerbach once attended a kiddush where no kiddush cup or even glass cup was available. Rav Shlomo Zalman did not hesitate to make kiddush using a plastic cup. When someone mentioned that Rav Moshe Feinstein said it was assur to use a throw away cup. Rav Shlomo Zalman did not flinch and said it is mutar beyond a shadow of a doubt. Rav Shlomo Zalman gave a Mashal to shirts that are made for one time use and sold in America. “If I had a shirt like that” said Rav Shlomo Zalman, “Do you think I would throw it away after one use?”Rav Shlomo Zalman’s son-in-law, Rav Zalman Nechemia Goldberg proved from the Rambam that any vessel meant for one time use, is not considered a Keili. However he says that it depends on the users intention. If does not plan on throwing it out, then it is a Keili, even if he is the only one in the whole world that reuses it.
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reb moishe disscusses paper specifically which is more disposable than plastic also other leading poskim in erez yisroel are lenient see shut tzitz eliezer
Some use two cups (together) so they don’t throw away the bottom cup.