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 ) |
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 ) |