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Mirrors > Home > ILE Home > Th. List > znnnlt1 | GIF version |
Description: An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.) |
Ref | Expression |
---|---|
znnnlt1 | ⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnnz1 8871 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
2 | 1 | baib 869 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℕ ↔ 1 ≤ 𝑁)) |
3 | 2 | notbid 630 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ ¬ 1 ≤ 𝑁)) |
4 | 1z 8874 | . . 3 ⊢ 1 ∈ ℤ | |
5 | zltnle 8894 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 < 1 ↔ ¬ 1 ≤ 𝑁)) | |
6 | 4, 5 | mpan2 417 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 < 1 ↔ ¬ 1 ≤ 𝑁)) |
7 | 3, 6 | bitr4d 190 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1445 class class class wbr 3867 1c1 7448 < clt 7619 ≤ cle 7620 ℕcn 8520 ℤcz 8848 |
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-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 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-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 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-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 |
This theorem is referenced by: (None) |
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