Step |
Hyp |
Ref |
Expression |
1 |
|
tskmval |
⊢ ( 𝐴 ∈ V → ( tarskiMap ‘ 𝐴 ) = ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ) |
2 |
|
ssrab2 |
⊢ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ⊆ Tarski |
3 |
|
id |
⊢ ( 𝐴 ∈ V → 𝐴 ∈ V ) |
4 |
|
grothtsk |
⊢ ∪ Tarski = V |
5 |
3 4
|
eleqtrrdi |
⊢ ( 𝐴 ∈ V → 𝐴 ∈ ∪ Tarski ) |
6 |
|
eluni2 |
⊢ ( 𝐴 ∈ ∪ Tarski ↔ ∃ 𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ) |
7 |
5 6
|
sylib |
⊢ ( 𝐴 ∈ V → ∃ 𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ) |
8 |
|
rabn0 |
⊢ ( { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ≠ ∅ ↔ ∃ 𝑥 ∈ Tarski 𝐴 ∈ 𝑥 ) |
9 |
7 8
|
sylibr |
⊢ ( 𝐴 ∈ V → { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ≠ ∅ ) |
10 |
|
inttsk |
⊢ ( ( { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ⊆ Tarski ∧ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ≠ ∅ ) → ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ∈ Tarski ) |
11 |
2 9 10
|
sylancr |
⊢ ( 𝐴 ∈ V → ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ∈ Tarski ) |
12 |
1 11
|
eqeltrd |
⊢ ( 𝐴 ∈ V → ( tarskiMap ‘ 𝐴 ) ∈ Tarski ) |
13 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( tarskiMap ‘ 𝐴 ) = ∅ ) |
14 |
|
0tsk |
⊢ ∅ ∈ Tarski |
15 |
13 14
|
eqeltrdi |
⊢ ( ¬ 𝐴 ∈ V → ( tarskiMap ‘ 𝐴 ) ∈ Tarski ) |
16 |
12 15
|
pm2.61i |
⊢ ( tarskiMap ‘ 𝐴 ) ∈ Tarski |