0mnnnnn0 - Metamath Proof Explorer

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Theorem 0mnnnnn0 11921

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 10635	. 2 ⊢ 0 ∈ ℝ
2		nnel 3130	. . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ₀ ↔ (0 − 𝑁) ∈ ℕ₀)
3		df-neg 10865	. . . . . 6 ⊢ -𝑁 = (0 − 𝑁)
4	3	eqcomi 2828	. . . . 5 ⊢ (0 − 𝑁) = -𝑁
5	4	eleq1i 2901	. . . 4 ⊢ ((0 − 𝑁) ∈ ℕ₀ ↔ -𝑁 ∈ ℕ₀)
6		nn0ge0 11914	. . . . 5 ⊢ (-𝑁 ∈ ℕ₀ → 0 ≤ -𝑁)
7		nnre 11637	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
8	7	le0neg1d 11203	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
9		nngt0 11660	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10		0red 10636	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
11	10, 7	ltnled 10779	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0))
12		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ))
13	11, 12	syl6bi 255	. . . . . . 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 262	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ))
16	6, 15	syl5 34	. . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ₀ → ¬ 0 ∈ ℝ))
17	5, 16	syl5bi 244	. . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ₀ → ¬ 0 ∈ ℝ))
18	2, 17	syl5bi 244	. 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 3121 class class class wbr 5057 (class class class)co 7148 ℝcr 10528 0cc0 10529 < clt 10667 ≤ cle 10668 − cmin 10862 -cneg 10863 ℕcn 11630 ℕ₀cn0 11889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890

This theorem is referenced by: 0nn0m1nnn0 32344

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