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Mirrors > Home > ILE Home > Th. List > 0mnnnnn0 | GIF version |
Description: The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
Ref | Expression |
---|---|
0mnnnnn0 | ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7988 | . . 3 ⊢ 0 ∈ ℝ | |
2 | df-neg 8162 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
3 | 2 | eqcomi 2193 | . . . . 5 ⊢ (0 − 𝑁) = -𝑁 |
4 | 3 | eleq1i 2255 | . . . 4 ⊢ ((0 − 𝑁) ∈ ℕ0 ↔ -𝑁 ∈ ℕ0) |
5 | nn0ge0 9232 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → 0 ≤ -𝑁) | |
6 | nnre 8957 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | 6 | le0neg1d 8505 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) |
8 | nngt0 8975 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
9 | 0red 7989 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ) | |
10 | 6, 9 | lenltd 8106 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁)) |
11 | pm2.21 618 | . . . . . . . 8 ⊢ (¬ 0 < 𝑁 → (0 < 𝑁 → ¬ 0 ∈ ℝ)) | |
12 | 10, 11 | biimtrdi 163 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 → (0 < 𝑁 → ¬ 0 ∈ ℝ))) |
13 | 8, 12 | mpid 42 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ)) |
14 | 7, 13 | sylbird 170 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ)) |
15 | 5, 14 | syl5 32 | . . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
16 | 4, 15 | biimtrid 152 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
17 | 1, 16 | mt2i 645 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (0 − 𝑁) ∈ ℕ0) |
18 | df-nel 2456 | . 2 ⊢ ((0 − 𝑁) ∉ ℕ0 ↔ ¬ (0 − 𝑁) ∈ ℕ0) | |
19 | 17, 18 | sylibr 134 | 1 ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2160 ∉ wnel 2455 class class class wbr 4018 (class class class)co 5897 ℝcr 7841 0cc0 7842 < clt 8023 ≤ cle 8024 − cmin 8159 -cneg 8160 ℕcn 8950 ℕ0cn0 9207 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 |
This theorem is referenced by: (None) |
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