0mnnnnn0 - Metamath Proof Explorer

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Theorem 0mnnnnn0 12008

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 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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