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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 7766 | . . 3 ⊢ 0 ∈ ℝ | |
2 | df-neg 7936 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
3 | 2 | eqcomi 2143 | . . . . 5 ⊢ (0 − 𝑁) = -𝑁 |
4 | 3 | eleq1i 2205 | . . . 4 ⊢ ((0 − 𝑁) ∈ ℕ0 ↔ -𝑁 ∈ ℕ0) |
5 | nn0ge0 9002 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → 0 ≤ -𝑁) | |
6 | nnre 8727 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | 6 | le0neg1d 8279 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) |
8 | nngt0 8745 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
9 | 0red 7767 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ) | |
10 | 6, 9 | lenltd 7880 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁)) |
11 | pm2.21 606 | . . . . . . . 8 ⊢ (¬ 0 < 𝑁 → (0 < 𝑁 → ¬ 0 ∈ ℝ)) | |
12 | 10, 11 | syl6bi 162 | . . . . . . 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 169 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ)) |
15 | 5, 14 | syl5 32 | . . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
16 | 4, 15 | syl5bi 151 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
17 | 1, 16 | mt2i 633 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (0 − 𝑁) ∈ ℕ0) |
18 | df-nel 2404 | . 2 ⊢ ((0 − 𝑁) ∉ ℕ0 ↔ ¬ (0 − 𝑁) ∈ ℕ0) | |
19 | 17, 18 | sylibr 133 | 1 ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1480 ∉ wnel 2403 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 < clt 7800 ≤ cle 7801 − cmin 7933 -cneg 7934 ℕcn 8720 ℕ0cn0 8977 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 |
This theorem is referenced by: (None) |
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