Exercise IV.17 in Kunen (1980): Absolute in all transitive models of a finite set of ZF axioms implies $\Delta_1$
137982
Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$
69337
Zorn's Lemma And Axiom of Choice
3663
Finding counterexamples in elementary set theory.
28452
Statement that is provable in $ZFC+CH$ yet unprovable in $ZFC+\lnot CH$
44259
Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)
56143
Proof of a basic $AC_\omega$ equivalence
78290
Countable axiom of choice: why you can't prove it from just ZF
28629
Is the axiom of choice necessary to prove that closed points in the Zariski topology are maximal ideals?
19604
Complete first order theory with finite model is categorical
35140
Existence of minimal $\sigma$-algebra and transfinite induction
153652
Countable ordinals are embeddable in the rationals $\Bbb Q$ -- proofs and their use of AC
9158
Proving that a propositional theory of any cardinality has an independent set of axioms
83185
How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?
75007
How can I prove this relation between the elementary set theory and the elementary logic?
83757
Is the following equivalent to the axiom of choice?
27899
Can we prove the existence of $A\cup B$ without the union axiom?
33563
Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)