0mnnnnn0 - Metamath Proof Explorer

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Theorem 0mnnnnn0 12195

Description: The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)

**Assertion**
Ref	Expression
0mnnnnn0	⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ₀)

**Proof of Theorem 0mnnnnn0**
Step	Hyp	Ref	Expression
1		0re 10908	. 2 ⊢ 0 ∈ ℝ
2		nnel 3057	. . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ₀ ↔ (0 − 𝑁) ∈ ℕ₀)
3		df-neg 11138	. . . . . 6 ⊢ -𝑁 = (0 − 𝑁)
4	3	eqcomi 2747	. . . . 5 ⊢ (0 − 𝑁) = -𝑁
5	4	eleq1i 2829	. . . 4 ⊢ ((0 − 𝑁) ∈ ℕ₀ ↔ -𝑁 ∈ ℕ₀)
6		nn0ge0 12188	. . . . 5 ⊢ (-𝑁 ∈ ℕ₀ → 0 ≤ -𝑁)
7		nnre 11910	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
8	7	le0neg1d 11476	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
9		nngt0 11934	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10		0red 10909	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
11	10, 7	ltnled 11052	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0))
12		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ))
13	11, 12	syl6bi 252	. . . . . . 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 259	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ))
16	6, 15	syl5 34	. . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ₀ → ¬ 0 ∈ ℝ))
17	5, 16	syl5bi 241	. . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ₀ → ¬ 0 ∈ ℝ))
18	2, 17	syl5bi 241	. 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 2108 ∉ wnel 3048 class class class wbr 5070 (class class class)co 7255 ℝcr 10801 0cc0 10802 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 ℕcn 11903 ℕ₀cn0 12163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164

This theorem is referenced by: 0nn0m1nnn0 32971

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