Subset Relation is Antisymmetric

Theorem

The subset relation is antisymmetric:

$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$

where $x$ and $y$ are sets.

Proof

This is a direct statement of the definition of set equality:

$x = y := \paren {x \subseteq y} \land \paren {y \subseteq x}$

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): symmetric relation
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): symmetric relation