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Mirrors > Home > MPE Home > Th. List > 0mnnnnn0 | Structured version Visualization version 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 10721 | . 2 ⊢ 0 ∈ ℝ | |
2 | nnel 3047 | . . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ0 ↔ (0 − 𝑁) ∈ ℕ0) | |
3 | df-neg 10951 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
4 | 3 | eqcomi 2747 | . . . . 5 ⊢ (0 − 𝑁) = -𝑁 |
5 | 4 | eleq1i 2823 | . . . 4 ⊢ ((0 − 𝑁) ∈ ℕ0 ↔ -𝑁 ∈ ℕ0) |
6 | nn0ge0 12001 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → 0 ≤ -𝑁) | |
7 | nnre 11723 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
8 | 7 | le0neg1d 11289 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) |
9 | nngt0 11747 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
10 | 0red 10722 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ) | |
11 | 10, 7 | ltnled 10865 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0)) |
12 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ)) | |
13 | 11, 12 | syl6bi 256 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ))) |
14 | 9, 13 | mpd 15 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ)) |
15 | 8, 14 | sylbird 263 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ)) |
16 | 6, 15 | syl5 34 | . . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
17 | 5, 16 | syl5bi 245 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
18 | 2, 17 | syl5bi 245 | . 2 ⊢ (𝑁 ∈ ℕ → (¬ (0 − 𝑁) ∉ ℕ0 → ¬ 0 ∈ ℝ)) |
19 | 1, 18 | mt4i 118 | 1 ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 ∉ wnel 3038 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 < clt 10753 ≤ cle 10754 − cmin 10948 -cneg 10949 ℕcn 11716 ℕ0cn0 11976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 |
This theorem is referenced by: 0nn0m1nnn0 32642 |
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