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Mirrors > Home > MPE Home > Th. List > 0mnnnnn0 | Structured version Visualization version GIF version |
Description: The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
Ref | Expression |
---|---|
0mnnnnn0 | ⊢ (𝑁 ∈ ℕ → (0 − 𝑁) ∉ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11158 | . 2 ⊢ 0 ∈ ℝ | |
2 | nnel 3059 | . . 3 ⊢ (¬ (0 − 𝑁) ∉ ℕ0 ↔ (0 − 𝑁) ∈ ℕ0) | |
3 | df-neg 11389 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
4 | 3 | eqcomi 2746 | . . . . 5 ⊢ (0 − 𝑁) = -𝑁 |
5 | 4 | eleq1i 2829 | . . . 4 ⊢ ((0 − 𝑁) ∈ ℕ0 ↔ -𝑁 ∈ ℕ0) |
6 | nn0ge0 12439 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → 0 ≤ -𝑁) | |
7 | nnre 12161 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
8 | 7 | le0neg1d 11727 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁)) |
9 | nngt0 12185 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
10 | 0red 11159 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℝ) | |
11 | 10, 7 | ltnled 11303 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 ↔ ¬ 𝑁 ≤ 0)) |
12 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 𝑁 ≤ 0 → (𝑁 ≤ 0 → ¬ 0 ∈ ℝ)) | |
13 | 11, 12 | syl6bi 253 | . . . . . . 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 260 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ -𝑁 → ¬ 0 ∈ ℝ)) |
16 | 6, 15 | syl5 34 | . . . 4 ⊢ (𝑁 ∈ ℕ → (-𝑁 ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
17 | 5, 16 | biimtrid 241 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 − 𝑁) ∈ ℕ0 → ¬ 0 ∈ ℝ)) |
18 | 2, 17 | biimtrid 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 2107 ∉ wnel 3050 class class class wbr 5106 (class class class)co 7358 ℝcr 11051 0cc0 11052 < clt 11190 ≤ cle 11191 − cmin 11386 -cneg 11387 ℕcn 12154 ℕ0cn0 12414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 |
This theorem is referenced by: 0nn0m1nnn0 33706 |
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