Zermelo-Fraenkel Axioms for Set Theory

The Zermelo-Fraenkel Axioms for Set Theory, or just `ZFC' for short, include the following nine axioms.

Axiom 7.1 (Axiom of Extensionality)  

$$\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y)$$

Axiom 7.2 (Axiom Schema of Separation)  

$$\forall x_0 \ldots \forall x_{n-1} \forall x \exists y \foral... ...\leftrightarrow (z \in x \wedge \phi(z, x_0, \ldots, x_{n-1})))$$

Axiom 7.3 (Pair Set Axiom)  

$$\forall x \forall y \exists z \forall w (w \in z \leftrightarrow (w = x \vee w = y))$$

Axiom 7.4 (Sum Set Axiom)  

$$\forall x \exists y \forall z (z \in y \leftrightarrow \exists w (w \in x \wedge z \in w))$$

Axiom 7.5 (Power Set Axiom)  

$$\forall x \exists y \forall z (z \in y \leftrightarrow \forall w (w \in z \rightarrow w \in x))$$

Axiom 7.6 (Axiom of Infinity)  

$$\exists x (\emptyset \in x \wedge \forall y (y \in x \rightarrow y \cup \{y\} \in x))$$

Axiom 7.7 (Axiom Schema of Replacement)  

$$\forall x_0 \ldots \forall x_{n-1} (\forall x \exists^{=1} y ... ...w \exists x (x \in u \wedge \phi(x, y, x_0, \ldots, x_{n-1}))))$$

Axiom 7.8 (Axiom of Choice)  

$$\forall x ((\emptyset \not\in x \wedge \forall u \forall v ((... ... \forall w (w \in x \rightarrow \exists^{=1} z z \in w \cap y))$$