Step |
Hyp |
Ref |
Expression |
1 |
|
snexg |
⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V ) |
2 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
3 |
|
eleq2 |
⊢ ( 𝑥 = { 𝐴 } → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ { 𝐴 } ) ) |
4 |
1 2 3
|
elabd |
⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ) |
5 |
|
intss1 |
⊢ ( { 𝐴 } ∈ { 𝑥 ∣ 𝐴 ∈ 𝑥 } → ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ⊆ { 𝐴 } ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ⊆ { 𝐴 } ) |
7 |
|
id |
⊢ ( 𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥 ) |
8 |
7
|
ax-gen |
⊢ ∀ 𝑥 ( 𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥 ) |
9 |
|
elintabg |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ( 𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥 ) ) ) |
10 |
8 9
|
mpbiri |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ) |
11 |
10
|
snssd |
⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ⊆ ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } ) |
12 |
6 11
|
eqssd |
⊢ ( 𝐴 ∈ 𝑉 → ∩ { 𝑥 ∣ 𝐴 ∈ 𝑥 } = { 𝐴 } ) |