Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ Singletons → 𝐴 ∈ V ) |
2 |
|
snex |
⊢ { 𝑥 } ∈ V |
3 |
|
eleq1 |
⊢ ( 𝐴 = { 𝑥 } → ( 𝐴 ∈ V ↔ { 𝑥 } ∈ V ) ) |
4 |
2 3
|
mpbiri |
⊢ ( 𝐴 = { 𝑥 } → 𝐴 ∈ V ) |
5 |
4
|
exlimiv |
⊢ ( ∃ 𝑥 𝐴 = { 𝑥 } → 𝐴 ∈ V ) |
6 |
|
eleq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ) ) |
7 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 = { 𝑥 } ↔ 𝐴 = { 𝑥 } ) ) |
8 |
7
|
exbidv |
⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 𝑦 = { 𝑥 } ↔ ∃ 𝑥 𝐴 = { 𝑥 } ) ) |
9 |
|
df-singles |
⊢ Singletons = ran Singleton |
10 |
9
|
eleq2i |
⊢ ( 𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton ) |
11 |
|
vex |
⊢ 𝑦 ∈ V |
12 |
11
|
elrn |
⊢ ( 𝑦 ∈ ran Singleton ↔ ∃ 𝑥 𝑥 Singleton 𝑦 ) |
13 |
|
vex |
⊢ 𝑥 ∈ V |
14 |
13 11
|
brsingle |
⊢ ( 𝑥 Singleton 𝑦 ↔ 𝑦 = { 𝑥 } ) |
15 |
14
|
exbii |
⊢ ( ∃ 𝑥 𝑥 Singleton 𝑦 ↔ ∃ 𝑥 𝑦 = { 𝑥 } ) |
16 |
10 12 15
|
3bitri |
⊢ ( 𝑦 ∈ Singletons ↔ ∃ 𝑥 𝑦 = { 𝑥 } ) |
17 |
6 8 16
|
vtoclbg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ Singletons ↔ ∃ 𝑥 𝐴 = { 𝑥 } ) ) |
18 |
1 5 17
|
pm5.21nii |
⊢ ( 𝐴 ∈ Singletons ↔ ∃ 𝑥 𝐴 = { 𝑥 } ) |