Metamath Proof Explorer


Theorem prmunb2

Description: The primes are unbounded. This generalizes prmunb to real A with arch and lttrd : every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020)

Ref Expression
Assertion prmunb2 ( 𝐴 ∈ ℝ → ∃ 𝑝 ∈ ℙ 𝐴 < 𝑝 )

Proof

Step Hyp Ref Expression
1 simplll ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛𝑛 < 𝑝 ) ) → 𝐴 ∈ ℝ )
2 nnre ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ )
3 2 ad3antlr ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛𝑛 < 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 < 𝑛𝑛 < 𝑝 ) ) → 𝑝 ∈ ℝ )
7 simprl ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛𝑛 < 𝑝 ) ) → 𝐴 < 𝑛 )
8 simprr ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛𝑛 < 𝑝 ) ) → 𝑛 < 𝑝 )
9 1 3 6 7 8 lttrd ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ ) ∧ 𝑝 ∈ ℙ ) ∧ ( 𝐴 < 𝑛𝑛 < 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 < 𝑛𝑛 < 𝑝 ) ↔ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) )
16 15 rexbii ( ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ ( 𝐴 < 𝑛𝑛 < 𝑝 ) ↔ ∃ 𝑛 ∈ ℕ ( 𝐴 < 𝑛 ∧ ∃ 𝑝 ∈ ℙ 𝑛 < 𝑝 ) )
17 14 16 sylibr ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ ∃ 𝑝 ∈ ℙ ( 𝐴 < 𝑛𝑛 < 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 )