Step |
Hyp |
Ref |
Expression |
1 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝐴 ∈ ℝ ) |
2 |
|
nnre |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ ) |
3 |
2
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝑛 ∈ ℝ ) |
4 |
|
prmz |
⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℤ ) |
5 |
4
|
zred |
⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℝ ) |
6 |
5
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝑝 ∈ ℝ ) |
7 |
|
simprl |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝐴 < 𝑛 ) |
8 |
|
simprr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝑛 < 𝑝 ) |
9 |
1 3 6 7 8
|
lttrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) → 𝐴 < 𝑝 ) |
10 |
|
arch |
⊢ ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ 𝐴 < 𝑛 ) |
11 |
|
prmunb |
⊢ ( 𝑛 ∈ ℕ → ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) |
12 |
11
|
rgen |
⊢ ∀ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 |
13 |
|
r19.29r |
⊢ ( ( ∃ 𝑛 ∈ ℕ 𝐴 < 𝑛 ∧ ∀ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) → ∃ 𝑛 ∈ ℕ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) ) |
14 |
10 12 13
|
sylancl |
⊢ ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) ) |
15 |
|
r19.42v |
⊢ ( ∃ 𝑝 ∈ ℙ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ↔ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) ) |
16 |
15
|
rexbii |
⊢ ( ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ↔ ∃ 𝑛 ∈ ℕ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) ) |
17 |
14 16
|
sylibr |
⊢ ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ ( 𝐴 < 𝑛 ∧ 𝑛 < 𝑝 ) ) |
18 |
9 17
|
reximddv2 |
⊢ ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ) |
19 |
|
1nn |
⊢ 1 ∈ ℕ |
20 |
|
ne0i |
⊢ ( 1 ∈ ℕ → ℕ ≠ ∅ ) |
21 |
|
r19.9rzv |
⊢ ( ℕ ≠ ∅ → ( ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ) ) |
22 |
19 20 21
|
mp2b |
⊢ ( ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ) |
23 |
18 22
|
sylibr |
⊢ ( 𝐴 ∈ ℝ → ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 ) |