| Step |
Hyp |
Ref |
Expression |
| 1 |
|
elin |
⊢ ( 𝑣 ∈ ( ( { 𝑤 } × 𝑤 ) ∩ 𝑦 ) ↔ ( 𝑣 ∈ ( { 𝑤 } × 𝑤 ) ∧ 𝑣 ∈ 𝑦 ) ) |
| 2 |
|
elxp |
⊢ ( 𝑣 ∈ ( { 𝑤 } × 𝑤 ) ↔ ∃ 𝑡 ∃ 𝑔 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 3 |
|
excom |
⊢ ( ∃ 𝑡 ∃ 𝑔 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ↔ ∃ 𝑔 ∃ 𝑡 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 4 |
2 3
|
bitri |
⊢ ( 𝑣 ∈ ( { 𝑤 } × 𝑤 ) ↔ ∃ 𝑔 ∃ 𝑡 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 5 |
4
|
anbi1i |
⊢ ( ( 𝑣 ∈ ( { 𝑤 } × 𝑤 ) ∧ 𝑣 ∈ 𝑦 ) ↔ ( ∃ 𝑔 ∃ 𝑡 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ) |
| 6 |
|
19.41vv |
⊢ ( ∃ 𝑔 ∃ 𝑡 ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ( ∃ 𝑔 ∃ 𝑡 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ) |
| 7 |
|
an32 |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑣 ∈ 𝑦 ) ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 8 |
|
eleq1 |
⊢ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ → ( 𝑣 ∈ 𝑦 ↔ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) |
| 9 |
8
|
pm5.32i |
⊢ ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑣 ∈ 𝑦 ) ↔ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) |
| 10 |
|
velsn |
⊢ ( 𝑡 ∈ { 𝑤 } ↔ 𝑡 = 𝑤 ) |
| 11 |
10
|
anbi1i |
⊢ ( ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ↔ ( 𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤 ) ) |
| 12 |
9 11
|
anbi12i |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑣 ∈ 𝑦 ) ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ↔ ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ∧ ( 𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 13 |
|
an4 |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ∧ ( 𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤 ) ) ↔ ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑡 = 𝑤 ) ∧ ( ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤 ) ) ) |
| 14 |
|
ancom |
⊢ ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑡 = 𝑤 ) ↔ ( 𝑡 = 𝑤 ∧ 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ) ) |
| 15 |
|
ancom |
⊢ ( ( ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤 ) ↔ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) |
| 16 |
14 15
|
anbi12i |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ 𝑡 = 𝑤 ) ∧ ( ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ∧ 𝑔 ∈ 𝑤 ) ) ↔ ( ( 𝑡 = 𝑤 ∧ 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ) ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 17 |
|
anass |
⊢ ( ( ( 𝑡 = 𝑤 ∧ 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ) ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ↔ ( 𝑡 = 𝑤 ∧ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ) |
| 18 |
13 16 17
|
3bitri |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ∧ ( 𝑡 = 𝑤 ∧ 𝑔 ∈ 𝑤 ) ) ↔ ( 𝑡 = 𝑤 ∧ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ) |
| 19 |
7 12 18
|
3bitri |
⊢ ( ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ( 𝑡 = 𝑤 ∧ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ) |
| 20 |
19
|
exbii |
⊢ ( ∃ 𝑡 ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ∃ 𝑡 ( 𝑡 = 𝑤 ∧ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ) |
| 21 |
|
opeq1 |
⊢ ( 𝑡 = 𝑤 → ⟨ 𝑡 , 𝑔 ⟩ = ⟨ 𝑤 , 𝑔 ⟩ ) |
| 22 |
21
|
eqeq2d |
⊢ ( 𝑡 = 𝑤 → ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ↔ 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ) ) |
| 23 |
21
|
eleq1d |
⊢ ( 𝑡 = 𝑤 → ( ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ↔ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) |
| 24 |
23
|
anbi2d |
⊢ ( 𝑡 = 𝑤 → ( ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ↔ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 25 |
22 24
|
anbi12d |
⊢ ( 𝑡 = 𝑤 → ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ↔ ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ) |
| 26 |
25
|
equsexvw |
⊢ ( ∃ 𝑡 ( 𝑡 = 𝑤 ∧ ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑡 , 𝑔 ⟩ ∈ 𝑦 ) ) ) ↔ ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 27 |
20 26
|
bitri |
⊢ ( ∃ 𝑡 ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 28 |
27
|
exbii |
⊢ ( ∃ 𝑔 ∃ 𝑡 ( ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ∃ 𝑔 ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 29 |
6 28
|
bitr3i |
⊢ ( ( ∃ 𝑔 ∃ 𝑡 ( 𝑣 = ⟨ 𝑡 , 𝑔 ⟩ ∧ ( 𝑡 ∈ { 𝑤 } ∧ 𝑔 ∈ 𝑤 ) ) ∧ 𝑣 ∈ 𝑦 ) ↔ ∃ 𝑔 ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 30 |
1 5 29
|
3bitri |
⊢ ( 𝑣 ∈ ( ( { 𝑤 } × 𝑤 ) ∩ 𝑦 ) ↔ ∃ 𝑔 ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 31 |
30
|
eubii |
⊢ ( ∃! 𝑣 𝑣 ∈ ( ( { 𝑤 } × 𝑤 ) ∩ 𝑦 ) ↔ ∃! 𝑣 ∃ 𝑔 ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ) |
| 32 |
|
vex |
⊢ 𝑤 ∈ V |
| 33 |
32
|
euop2 |
⊢ ( ∃! 𝑣 ∃ 𝑔 ( 𝑣 = ⟨ 𝑤 , 𝑔 ⟩ ∧ ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) ↔ ∃! 𝑔 ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) |
| 34 |
31 33
|
bitri |
⊢ ( ∃! 𝑣 𝑣 ∈ ( ( { 𝑤 } × 𝑤 ) ∩ 𝑦 ) ↔ ∃! 𝑔 ( 𝑔 ∈ 𝑤 ∧ ⟨ 𝑤 , 𝑔 ⟩ ∈ 𝑦 ) ) |