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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 7790 | . . 3 ⊢ 0 ∈ ℝ | |
2 | df-neg 7960 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
3 | 2 | eqcomi 2144 | . . . . 5 ⊢ (0 − 𝑁) = -𝑁 |
4 | 3 | eleq1i 2206 | . . . 4 ⊢ ((0 − 𝑁) ∈ ℕ0 ↔ -𝑁 ∈ ℕ0) |
5 | nn0ge0 9026 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → 0 ≤ -𝑁) | |
6 | nnre 8751 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
7 | 6 | le0neg1d 8303 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) |
8 | nngt0 8769 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
9 | 0red 7791 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ) | |
10 | 6, 9 | lenltd 7904 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁)) |
11 | pm2.21 607 | . . . . . . . 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 634 | . 2 ⊢ (𝑁 ∈ ℕ → ¬ (0 − 𝑁) ∈ ℕ0) |
18 | df-nel 2405 | . 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 1481 ∉ wnel 2404 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 < clt 7824 ≤ cle 7825 − cmin 7957 -cneg 7958 ℕcn 8744 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 |
This theorem is referenced by: (None) |
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