では is a collocation of で and は. で is the ren'yōkei(連用形)/-masu stem of the copula だ. Note that there is a case particle で (e.g. 電車で行きます), but this で is not a particle.
は is indeed a particle, but a kakari-joshi(係助詞)/binding particle, not a case particle like が, を, に or の.
じゃありません/ではありません is a variation of でありません, and でありません is the polite negative form of である. Thus, じゃありません/ではありません is a polite negative form of だ.
On the other hand, ありません is the negative form of あります, and あります is the polite form of the verb ある. The negative form of ある is ない, thus, ありません is a polite form of ない.
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