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Trvially, the empty set $\emptyset$ is a subset of every set $V$(say). Does it mean that $\emptyset$ is also a subset of every subset of $V$? i.e , if $A$ is an arbitrary subset of $V$, then will it be right to say that $\emptyset$ $\subseteq A$?

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marked as duplicate by Asaf Karagila elementary-set-theory Feb 4 at 17:14

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    $\begingroup$ Every bag can be made empty by removing things from it, but not every bag literally has an empty bag inside it. (The bags analogy is only helpful up to a point, but I think it is helpful here.) $\endgroup$ – Noah Schweber Feb 4 at 17:14
  • $\begingroup$ By "belongs to" -- do you mean "is an element of" or "is a subset of"? $\endgroup$ – Jakob B. Feb 4 at 17:16
  • $\begingroup$ Probably a handful of other questions are also good as duplicates. $\endgroup$ – Asaf Karagila Feb 4 at 17:17
  • $\begingroup$ If a statement is true for all sets, and a subset of a set is in particular a set (it is a sub-set), then the statement is true for all subsets of a given set as well, yes. $\endgroup$ – Asaf Karagila Feb 4 at 17:44
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No: $\emptyset \notin \emptyset$ (but $\emptyset \subseteq \emptyset$).

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  • $\begingroup$ do you mean that empty set is a subet of every subset? If so, i will be done $\endgroup$ – gete Feb 4 at 17:13
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    $\begingroup$ @gete Yes - a subset, but not necessarily an element. $\endgroup$ – Noah Schweber Feb 4 at 17:16
  • $\begingroup$ @gete I interpreted "belongs to" as "is an element of". You might want to update your question, if you meant something else $\endgroup$ – Jakob B. Feb 4 at 17:18
  • $\begingroup$ @bruderjakob17 my question was a bit wrongly framed anyways, i think Noah Schweber has got the answer in his comment. I am editing my pose. $\endgroup$ – gete Feb 4 at 17:27

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