| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vex 3483 | . . . . . . . . 9
⊢ 𝑦 ∈ V | 
| 2 | 1 | elpr 4649 | . . . . . . . 8
⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) | 
| 3 | 2 | anbi2i 623 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))) | 
| 4 |  | ancom 460 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦)) | 
| 5 |  | andir 1010 | . . . . . . . 8
⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) ∧ 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 6 | 4, 5 | bitri 275 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 7 | 3, 6 | bitri 275 | . . . . . 6
⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 8 | 7 | exbii 1847 | . . . . 5
⊢
(∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 9 |  | 19.43 1881 | . . . . 5
⊢
(∃𝑦((𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 10 | 8, 9 | bitri 275 | . . . 4
⊢
(∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 11 |  | clel3g 3660 | . . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))) | 
| 12 | 11 | bicomd 223 | . . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴)) | 
| 13 | 12 | adantr 480 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐴)) | 
| 14 |  | clel3g 3660 | . . . . . . 7
⊢ (𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))) | 
| 15 | 14 | bicomd 223 | . . . . . 6
⊢ (𝐵 ∈ 𝑊 → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵)) | 
| 16 | 15 | adantl 481 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ 𝑥 ∈ 𝐵)) | 
| 17 | 13, 16 | orbi12d 918 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ∨ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))) | 
| 18 | 10, 17 | bitrid 283 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))) | 
| 19 | 18 | abbidv 2807 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}) | 
| 20 |  | df-uni 4907 | . 2
⊢ ∪ {𝐴,
𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})} | 
| 21 |  | df-un 3955 | . 2
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} | 
| 22 | 19, 20, 21 | 3eqtr4g 2801 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |