Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | bnj1138.1 | ⊢ 𝐴 = ( 𝐵 ∪ 𝐶 ) | |
Assertion | bnj1138 | ⊢ ( 𝑋 ∈ 𝐴 ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1138.1 | ⊢ 𝐴 = ( 𝐵 ∪ 𝐶 ) | |
2 | 1 | eleq2i | ⊢ ( 𝑋 ∈ 𝐴 ↔ 𝑋 ∈ ( 𝐵 ∪ 𝐶 ) ) |
3 | elun | ⊢ ( 𝑋 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) ) | |
4 | 2 3 | bitri | ⊢ ( 𝑋 ∈ 𝐴 ↔ ( 𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶 ) ) |