Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) → 𝐴 ∈ V ) |
2 |
|
elex |
⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V ) |
3 |
|
elex |
⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V ) |
4 |
2 3
|
jaoi |
⊢ ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) → 𝐴 ∈ V ) |
5 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) ) |
6 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶 ) ) |
7 |
5 6
|
orbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ) ) |
8 |
|
eleq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) ) |
9 |
|
eleq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) ) |
10 |
8 9
|
orbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) ) ) |
11 |
|
df-un |
⊢ ( 𝐵 ∪ 𝐶 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) } |
12 |
7 10 11
|
elab2gw |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) ) ) |
13 |
1 4 12
|
pm5.21nii |
⊢ ( 𝐴 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶 ) ) |