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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 8543 | . . 3 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1 | biantrud 299 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) |
3 | nnre 8527 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | 1re 7584 | . . 3 ⊢ 1 ∈ ℝ | |
5 | letri3 7663 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 = 1 ↔ (𝐴 ≤ 1 ∧ 1 ≤ 𝐴))) | |
6 | 3, 4, 5 | sylancl 405 | . 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 1296 ∈ wcel 1445 class class class wbr 3867 ℝcr 7446 1c1 7448 ≤ cle 7620 ℕcn 8520 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1re 7536 ax-addrcl 7539 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-pre-ltirr 7554 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-xp 4473 df-cnv 4475 df-iota 5014 df-fv 5057 df-ov 5693 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-inn 8521 |
This theorem is referenced by: gcd1 11421 bezoutr1 11465 rpdvds 11524 isprm6 11569 qden1elz 11626 phimullem 11644 |
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