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We also observe that if V<\/jats:italic> is a model of $$\\textsf {ZFC}$$<\/jats:tex-math>\n ZFC<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then $$\\textsf {DC}_{<\\kappa }$$<\/jats:tex-math>\n \n DC<\/mml:mi>\n \n <<\/mml:mo>\n \u03ba<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be preserved in the symmetric extension of V<\/jats:italic> in terms of symmetric system $$\\langle {\\mathbb {P}},{\\mathcal {G}},{\\mathcal {F}}\\rangle $$<\/jats:tex-math>\n \n \u27e8<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n F<\/mml:mi>\n \u27e9<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if $${\\mathbb {P}}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$\\kappa $$<\/jats:tex-math>\n \u03ba<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-distributive and $${\\mathcal {F}}$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$\\kappa $$<\/jats:tex-math>\n \u03ba<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-complete. Further we observe that if $$\\delta <\\kappa $$<\/jats:tex-math>\n \n \u03b4<\/mml:mi>\n <<\/mml:mo>\n \u03ba<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and V<\/jats:italic> is a model of $$\\textsf {ZF}+\\textsf {DC}_{\\delta }$$<\/jats:tex-math>\n \n ZF<\/mml:mi>\n +<\/mml:mo>\n \n DC<\/mml:mi>\n \u03b4<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then $$\\textsf {DC}_{\\delta }$$<\/jats:tex-math>\n \n DC<\/mml:mi>\n \u03b4<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be preserved in the symmetric extension of V<\/jats:italic> in terms of symmetric system $$\\langle {\\mathbb {P}},{\\mathcal {G}},{\\mathcal {F}}\\rangle $$<\/jats:tex-math>\n \n \u27e8<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n F<\/mml:mi>\n \u27e9<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if $${\\mathbb {P}}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is ($$\\delta +1$$<\/jats:tex-math>\n \n \u03b4<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>)-strategically closed and $${\\mathcal {F}}$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is $$\\kappa $$<\/jats:tex-math>\n \u03ba<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-complete.<\/jats:p>","DOI":"10.1007\/s00153-022-00845-3","type":"journal-article","created":{"date-parts":[[2022,9,10]],"date-time":"2022-09-10T18:02:37Z","timestamp":1662832957000},"page":"369-399","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Combinatorial properties and dependent choice in symmetric extensions based on L\u00e9vy collapse"],"prefix":"10.1007","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4156-7209","authenticated-orcid":false,"given":"Amitayu","family":"Banerjee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,10]]},"reference":[{"issue":"2","key":"845_CR1","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1215\/00294527-1960434","volume":"54","author":"A Apter","year":"2013","unstructured":"Apter, A., Cody, B.: Consecutive singular cardinals and the continuum function. 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