Is there a way of proving that if $A$ is an infinite set and $x\not\in A$ then $|A|=|A\cup\{x\}|$ without relying on the axiom of choice.
31509
Is axiom of choice necessary for proving that every infinite set has a countably infinite subset?
134071
If A is a finite set and B is an infinite set, then B\A is an infinite set
14713
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
5313
Existence of how many sets is asserted by the axiom of choice in this case?
65308
Axiom of Choice: Family of non-empty sets mutually disjoint or not?
107383
Derived sets - prove $(A \cup B)' = A' \cup B'$
123979
Checking proof for: It is possible to find addition assumptions on the sets $A$ and $B$, so $P(A)\cup P(B) = P(A\cup B)$.
121847
Don't we need the axiom of choice to choose from a non-empty set?
62208
If $A$ is a non-empty set, then $A \notin A$
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
132868
Bijection between an infinite set and its union of a countably infinite set
49220
Open Sets of $\mathbb{R}^1$ and axiom of choice
145428
Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
88099
Not $\sigma$-compact set without axiom of choice
49709
Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous
66398
Union of a finite set and a countably infinite set is countably infinite