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 ∪ 𝑦 ) ∈ V ↔ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elun	⊢ ( 𝑡 ∈ ( 𝑥 ∪ 𝑦 ) ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) )
2	1	bj-clex	⊢ ( ( 𝑥 ∪ 𝑦 ) ∈ V ↔ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) ) )