Assume for any arbitrary sets $A,B$ if there is a onto map from $A\to B$, then $\exists$ a one-one map from $B\to A$. Prove the Axiom of Choice.
40586
There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.
43068
Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union C.
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
93040
$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki
130865
Why is the axiom of choice separated from the other axioms?
102202
The smallest infinity and the axiom of choice
45623
Axiom of Choice: What exactly is a choice, and when and why is it needed?
152272
Can you explain the "Axiom of choice" in simple terms?
19414
Uncountable subset with uncountable complement, without the Axiom of Choice
20252
The Axiom of Choice and the Cartesian Product.
131983
Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
77546
Does a "backwards" choice function imply the Axiom of Choice?
121847
Don't we need the axiom of choice to choose from a non-empty set?
10909
$|A \cup B| = \mathfrak{c}$, then $A$ or $B$ has cardinality $\mathfrak{c}$.
83714
$A \in B$ vs. $A \subset B$ for proofs
136835
ZF Set Theory - If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$
81509
The cardinality of a countable union of countable sets, without the axiom of choice