0mnnnnn0 - Metamath Proof Explorer

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Theorem 0mnnnnn0 11325

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 10040	. 2 ⊢ 0 ∈ ℝ
2		nnel 2906	. . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ₀ ↔ (0 − 𝑁) ∈ ℕ₀)
3		df-neg 10269	. . . . . 6 ⊢ -𝑁 = (0 − 𝑁)
4	3	eqcomi 2631	. . . . 5 ⊢ (0 − 𝑁) = -𝑁
5	4	eleq1i 2692	. . . 4 ⊢ ((0 − 𝑁) ∈ ℕ₀ ↔ -𝑁 ∈ ℕ₀)
6		nn0ge0 11318	. . . . 5 ⊢ (-𝑁 ∈ ℕ₀ → 0 ≤ -𝑁)
7		nnre 11027	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
8	7	le0neg1d 10599	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
9		nngt0 11049	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10		0red 10041	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
11	10, 7	ltnled 10184	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0))
12		pm2.21 120	. . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ))
13	11, 12	syl6bi 243	. . . . . . 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 250	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ))
16	6, 15	syl5 34	. . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ₀ → ¬ 0 ∈ ℝ))
17	5, 16	syl5bi 232	. . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ₀ → ¬ 0 ∈ ℝ))
18	2, 17	syl5bi 232	. 2 ⊢ (𝑁 ∈ ℕ → (¬ (0 − 𝑁) ∉ ℕ₀ → ¬ 0 ∈ ℝ))
19	1, 18	mt4i 153	1 ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ₀)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1990 ∉ wnel 2897 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 ℕcn 11020 ℕ₀cn0 11292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293

This theorem is referenced by: (None)

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