![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-nnne0 | Structured version Visualization version GIF version |
Description: nnne0 12242 without ax-mulcom 11170. (Contributed by SN, 25-Jan-2025.) |
Ref | Expression |
---|---|
sn-nnne0 | ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ne1 12279 | . . 3 ⊢ 0 ≠ 1 | |
2 | 0re 11212 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 11210 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | lttri2i 11324 | . . 3 ⊢ (0 ≠ 1 ↔ (0 < 1 ∨ 1 < 0)) |
5 | 1, 4 | mpbi 229 | . 2 ⊢ (0 < 1 ∨ 1 < 0) |
6 | breq2 5151 | . . . . . 6 ⊢ (𝑥 = 1 → (0 < 𝑥 ↔ 0 < 1)) | |
7 | breq2 5151 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (0 < 𝑥 ↔ 0 < 𝑦)) | |
8 | breq2 5151 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (0 < 𝑥 ↔ 0 < (𝑦 + 1))) | |
9 | breq2 5151 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
10 | id 22 | . . . . . 6 ⊢ (0 < 1 → 0 < 1) | |
11 | nnre 12215 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
12 | 11 | ad2antlr 725 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ) |
13 | 1red 11211 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 1 ∈ ℝ) | |
14 | simpr 485 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < 𝑦) | |
15 | simpll 765 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < 1) | |
16 | 12, 13, 14, 15 | sn-addgt0d 41316 | . . . . . 6 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < (𝑦 + 1)) |
17 | 6, 7, 8, 9, 10, 16 | nnindd 12228 | . . . . 5 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 0 < 𝐴) |
18 | 17 | gt0ne0d 11774 | . . . 4 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
19 | 18 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 0 < 1) → 𝐴 ≠ 0) |
20 | breq1 5150 | . . . . . 6 ⊢ (𝑥 = 1 → (𝑥 < 0 ↔ 1 < 0)) | |
21 | breq1 5150 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 < 0 ↔ 𝑦 < 0)) | |
22 | breq1 5150 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 < 0 ↔ (𝑦 + 1) < 0)) | |
23 | breq1 5150 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
24 | id 22 | . . . . . 6 ⊢ (1 < 0 → 1 < 0) | |
25 | 11 | ad2antlr 725 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 𝑦 ∈ ℝ) |
26 | 1red 11211 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 1 ∈ ℝ) | |
27 | simpr 485 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 𝑦 < 0) | |
28 | simpll 765 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 1 < 0) | |
29 | 25, 26, 27, 28 | sn-addlt0d 41315 | . . . . . 6 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → (𝑦 + 1) < 0) |
30 | 20, 21, 22, 23, 24, 29 | nnindd 12228 | . . . . 5 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 < 0) |
31 | 30 | lt0ne0d 11775 | . . . 4 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
32 | 31 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 1 < 0) → 𝐴 ≠ 0) |
33 | 19, 32 | jaodan 956 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ (0 < 1 ∨ 1 < 0)) → 𝐴 ≠ 0) |
34 | 5, 33 | mpan2 689 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ℕcn 12208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-nn 12209 df-2 12271 df-3 12272 df-resub 41235 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |