Theorem bj-axbun

Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun ). (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axbun	$$⊢x\cup y\in \mathrm{V}\leftrightarrow \exists z\forall t\left(t\in z\leftrightarrow \left(t\in x\vee t\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elun	$$⊢t\in \left(x\cup y\right)\leftrightarrow \left(t\in x\vee t\in y\right)$$
2	1	bj-clex	$$⊢x\cup y\in \mathrm{V}\leftrightarrow \exists z\forall t\left(t\in z\leftrightarrow \left(t\in x\vee t\in y\right)\right)$$