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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 8956 | . . 3 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1 | biantrud 304 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) |
3 | nnre 8940 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | 1re 7970 | . . 3 ⊢ 1 ∈ ℝ | |
5 | letri3 8052 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 = 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) | |
6 | 3, 4, 5 | sylancl 413 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) |
7 | 2, 6 | bitr4d 191 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 ℝcr 7824 1c1 7826 ≤ cle 8007 ℕcn 8933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1re 7919 ax-addrcl 7922 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-xp 4644 df-cnv 4646 df-iota 5190 df-fv 5236 df-ov 5891 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-inn 8934 |
This theorem is referenced by: gcd1 12002 bezoutr1 12048 rpdvds 12113 isprm6 12161 qden1elz 12219 phimullem 12239 pockthlem 12368 zabsle1 14696 2sqlem8a 14765 2sqlem8 14766 |
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