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Mirrors > Home > MPE Home > Th. List > isprm | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
isprm | ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4970 | . . . 4 ⊢ (𝑝 = 𝑃 → (𝑛 ∥ 𝑝 ↔ 𝑛 ∥ 𝑃)) | |
2 | 1 | rabbidv 3425 | . . 3 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} = {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃}) |
3 | 2 | breq1d 4976 | . 2 ⊢ (𝑝 = 𝑃 → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o)) |
4 | df-prm 15850 | . 2 ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o} | |
5 | 3, 4 | elrab2 3622 | 1 ⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {crab 3109 class class class wbr 4966 2oc2o 7952 ≈ cen 8359 ℕcn 11491 ∥ cdvds 15445 ℙcprime 15849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rab 3114 df-v 3439 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-br 4967 df-prm 15850 |
This theorem is referenced by: prmnn 15852 1nprm 15857 isprm2 15860 |
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