Theorem unexg

Description: The union of two sets is a set. Corollary 5.8 of TakeutiZaring p. 16. (Contributed by NM, 18-Sep-2006) Prove unexg first and then unex and unexb from it. (Revised by BJ, 21-Jul-2025)

	Ref	Expression
Assertion	unexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		uniprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) )
2		prex	⊢ { 𝐴 , 𝐵 } ∈ V
3	2	a1i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { 𝐴 , 𝐵 } ∈ V )
4	3	uniexd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∪ { 𝐴 , 𝐵 } ∈ V )
5	1 4	eqeltrrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )