Theorem isprm 16612

Description: The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
isprm	⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))

Distinct variable group:   𝑃,𝑛

Proof of Theorem isprm

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5104	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑛 ∥ 𝑝 ↔ 𝑛 ∥ 𝑃))
2	1	rabbidv 3408	. . 3 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃})
3	2	breq1d 5110	. 2 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))
4		df-prm 16611	. 2 ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o}
5	3, 4	elrab2 3651	1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401   class class class wbr 5100  2_oc2o 8401   ≈ cen 8892  ℕcn 12157   ∥ cdvds 16191  ℙcprime 16610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-prm 16611

This theorem is referenced by:  prmnn  16613  1nprm  16618  isprm2  16621