Step |
Hyp |
Ref |
Expression |
1 |
|
nnex |
⊢ ℕ ∈ V |
2 |
1
|
canth2 |
⊢ ℕ ≺ 𝒫 ℕ |
3 |
|
domnsym |
⊢ ( 𝒫 ℕ ≼ ℕ → ¬ ℕ ≺ 𝒫 ℕ ) |
4 |
2 3
|
mt2 |
⊢ ¬ 𝒫 ℕ ≼ ℕ |
5 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
6 |
|
simpl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → 𝐴 ⊆ ℝ ) |
7 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
8 |
7
|
ntropn |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ 𝐴 ⊆ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ∈ ( topGen ‘ ran (,) ) ) |
9 |
5 6 8
|
sylancr |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ∈ ( topGen ‘ ran (,) ) ) |
10 |
|
opnreen |
⊢ ( ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ∈ ( topGen ‘ ran (,) ) ∧ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≠ ∅ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≈ 𝒫 ℕ ) |
11 |
10
|
ex |
⊢ ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ∈ ( topGen ‘ ran (,) ) → ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≠ ∅ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≈ 𝒫 ℕ ) ) |
12 |
9 11
|
syl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≠ ∅ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≈ 𝒫 ℕ ) ) |
13 |
|
reex |
⊢ ℝ ∈ V |
14 |
13
|
ssex |
⊢ ( 𝐴 ⊆ ℝ → 𝐴 ∈ V ) |
15 |
7
|
ntrss2 |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ 𝐴 ⊆ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ⊆ 𝐴 ) |
16 |
5 15
|
mpan |
⊢ ( 𝐴 ⊆ ℝ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ⊆ 𝐴 ) |
17 |
|
ssdomg |
⊢ ( 𝐴 ∈ V → ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ⊆ 𝐴 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ 𝐴 ) ) |
18 |
14 16 17
|
sylc |
⊢ ( 𝐴 ⊆ ℝ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ 𝐴 ) |
19 |
|
domtr |
⊢ ( ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ 𝐴 ∧ 𝐴 ≼ ℕ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ ℕ ) |
20 |
18 19
|
sylan |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ ℕ ) |
21 |
|
ensym |
⊢ ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≈ 𝒫 ℕ → 𝒫 ℕ ≈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ) |
22 |
|
endomtr |
⊢ ( ( 𝒫 ℕ ≈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ∧ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ ℕ ) → 𝒫 ℕ ≼ ℕ ) |
23 |
22
|
expcom |
⊢ ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≼ ℕ → ( 𝒫 ℕ ≈ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) → 𝒫 ℕ ≼ ℕ ) ) |
24 |
20 21 23
|
syl2im |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≈ 𝒫 ℕ → 𝒫 ℕ ≼ ℕ ) ) |
25 |
12 24
|
syld |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) ≠ ∅ → 𝒫 ℕ ≼ ℕ ) ) |
26 |
25
|
necon1bd |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ¬ 𝒫 ℕ ≼ ℕ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) = ∅ ) ) |
27 |
4 26
|
mpi |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ 𝐴 ) = ∅ ) |