Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ ∪ 𝐵 → 𝐴 ∈ V ) |
2 |
|
elex |
⊢ ( 𝐴 ∈ 𝑥 → 𝐴 ∈ V ) |
3 |
2
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ V ) |
4 |
3
|
exlimiv |
⊢ ( ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) → 𝐴 ∈ V ) |
5 |
|
elequ1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) ) |
6 |
5
|
anbi1d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
7 |
6
|
exbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ∃ 𝑥 ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
8 |
|
eleq1 |
⊢ ( 𝑧 = 𝐴 → ( 𝑧 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥 ) ) |
9 |
8
|
anbi1d |
⊢ ( 𝑧 = 𝐴 → ( ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
10 |
9
|
exbidv |
⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑥 ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
11 |
|
df-uni |
⊢ ∪ 𝐵 = { 𝑦 ∣ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) } |
12 |
7 10 11
|
elab2gw |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
13 |
1 4 12
|
pm5.21nii |
⊢ ( 𝐴 ∈ ∪ 𝐵 ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) |