# Is empty set a subset of every subset? [duplicate]

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 $$\emptyset$$ $$\subseteq A$$?

## marked as duplicate by Asaf Karagila♦ elementary-set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Feb 4 at 17:14

• 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.) – Noah Schweber Feb 4 at 17:14
• By "belongs to" -- do you mean "is an element of" or "is a subset of"? – Jakob B. Feb 4 at 17:16
• Probably a handful of other questions are also good as duplicates. – Asaf Karagila Feb 4 at 17:17
• 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. – Asaf Karagila Feb 4 at 17:44

No: $$\emptyset \notin \emptyset$$ (but $$\emptyset \subseteq \emptyset$$).