0mnnnnn0 - Metamath Proof Explorer

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Theorem 0mnnnnn0 11783

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 10496	. 2 ⊢ 0 ∈ ℝ
2		nnel 3101	. . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ₀ ↔ (0 − 𝑁) ∈ ℕ₀)
3		df-neg 10726	. . . . . 6 ⊢ -𝑁 = (0 − 𝑁)
4	3	eqcomi 2806	. . . . 5 ⊢ (0 − 𝑁) = -𝑁
5	4	eleq1i 2875	. . . 4 ⊢ ((0 − 𝑁) ∈ ℕ₀ ↔ -𝑁 ∈ ℕ₀)
6		nn0ge0 11776	. . . . 5 ⊢ (-𝑁 ∈ ℕ₀ → 0 ≤ -𝑁)
7		nnre 11499	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
8	7	le0neg1d 11065	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
9		nngt0 11522	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
10		0red 10497	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ)
11	10, 7	ltnled 10640	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0))
12		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ))
13	11, 12	syl6bi 254	. . . . . . 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 261	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ))
16	6, 15	syl5 34	. . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ₀ → ¬ 0 ∈ ℝ))
17	5, 16	syl5bi 243	. . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ₀ → ¬ 0 ∈ ℝ))
18	2, 17	syl5bi 243	. 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 2083 ∉ wnel 3092 class class class wbr 4968 (class class class)co 7023 ℝcr 10389 0cc0 10390 < clt 10528 ≤ cle 10529 − cmin 10723 -cneg 10724 ℕcn 11492 ℕ₀cn0 11751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752

This theorem is referenced by: 0nn0m1nnn0 31961

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