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Mirrors > Home > ILE Home > Th. List > nnle1eq1 | GIF version |
Description: A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Ref | Expression |
---|---|
nnle1eq1 | ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8743 | . . 3 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1 | biantrud 302 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) |
3 | nnre 8727 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | 1re 7765 | . . 3 ⊢ 1 ∈ ℝ | |
5 | letri3 7845 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 = 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) | |
6 | 3, 4, 5 | sylancl 409 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) |
7 | 2, 6 | bitr4d 190 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 1c1 7621 ≤ cle 7801 ℕcn 8720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 |
This theorem is referenced by: gcd1 11675 bezoutr1 11721 rpdvds 11780 isprm6 11825 qden1elz 11883 phimullem 11901 |
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