Let A be an infinite set. Show that there exists a proper subset B of A such that |A| = |B|. Does your proof use the Axiom of Choice?
36281
A set is infinite iff it is equivalent to a proper subset of itself
79697
If $A$ is a subset of $B$, then the closure of $A$ is contained in the closure of $B$.
151295
Proving $|S \times \mathbb{N}|=|S|$ when $S$ is an infinite set
20706
Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)
10472
Is every set a subset?
70940
Homework - set theory infinite union
156006
Cardinality of set containing all infinite subsets of $\mathbb{Q}$
134914
Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
119624
A Dedekind infinite set has a countably infinite proper subset
80827
Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
130998
Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
83757
Is the following equivalent to the axiom of choice?
102202
The smallest infinity and the axiom of choice
144787
Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
153935
A weaker Axiom of Infinity?
12436
The cardinality of the set of all finite subsets of an infinite set
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?