והא תניא
*But it is taught*
Introduces a tannaitic text as the source of an objection.
והאנן… תנן
*But haven’t we learned… (in our mishnah)?*
Introduces a contradiction between a mishnah (usually the one currently under discussion) and a source/halakhah that has just been cited; the relevant part of the mishnah will be cited after or inbetween.