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  "body": "\n## Definition\n\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nThe set $A$ is a subset of $B$, written $A \\subseteq B$, when each [[element|0EZ758XE73]] of $A$ is an element of $B$:\n$$\\forall x \\, (x \\in A \\to x \\in B).$$\nThen $B$ is a superset of $A$.\nThe set $A$ is a proper subset of $B$, written $A \\subsetneq B$, when $A \\subseteq B$ and $A \\neq B$.\n\n## Remarks\n\nSome authors write $A \\subset B$ for $A \\subseteq B$, and others write it for a proper subset.\nTwo sets are equal if and only if each is a subset of the other. This follows from the [[axiom of extensionality|0TB16T2C97]].\nThe [[empty set|0ECVK59Y03]] is a subset of every set, and every set is a subset of itself.\n\n## Explanations\n\n### L1 human\n\n$A \\subseteq B$ means that $A$ fits inside $B$.\nThe set of even numbers is a subset of the set of integers.\nTo prove $A \\subseteq B$, take an arbitrary $x \\in A$ and show $x \\in B$.\n",
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