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Mirrors > Home > ILE Home > Th. List > nnm1ge0 | GIF version |
Description: A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.) |
Ref | Expression |
---|---|
nnm1ge0 | ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nngt0 8890 | . 2 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
2 | 0z 9210 | . . 3 ⊢ 0 ∈ ℤ | |
3 | nnz 9218 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
4 | zltlem1 9256 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 ↔ 0 ≤ (𝑁 − 1))) | |
5 | 2, 3, 4 | sylancr 412 | . 2 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ 0 ≤ (𝑁 − 1))) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 0cc0 7761 1c1 7762 < clt 7941 ≤ cle 7942 − cmin 8077 ℕcn 8865 ℤcz 9199 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 |
This theorem is referenced by: ubmelm1fzo 10169 m1modnnsub1 10313 uzwodc 11979 |
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