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Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$? [duplicate]

Does there exist a cardinal $\kappa$ such that $\aleph_{\kappa} = \kappa$ ? Moreover is there one that is regular? Thanks in advance!
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2answers
42 views

Image of a proper class under one-one function is a proper class

Working under Zermelo-Franenkel set theory. how I can prove following: Let $V$ be proper class. Let $F : V \rightarrow U$ be a one-one function. Then $U$ has to be a proper class.