![]() |
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 12298 without ax-mulcom 11217. (Contributed by SN, 25-Jan-2025.) |
Ref | Expression |
---|---|
sn-nnne0 | ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ne1 12335 | . . 3 ⊢ 0 ≠ 1 | |
2 | 0re 11261 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 11259 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | lttri2i 11373 | . . 3 ⊢ (0 ≠ 1 ↔ (0 < 1 ∨ 1 < 0)) |
5 | 1, 4 | mpbi 230 | . 2 ⊢ (0 < 1 ∨ 1 < 0) |
6 | breq2 5152 | . . . . . 6 ⊢ (𝑥 = 1 → (0 < 𝑥 ↔ 0 < 1)) | |
7 | breq2 5152 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (0 < 𝑥 ↔ 0 < 𝑦)) | |
8 | breq2 5152 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (0 < 𝑥 ↔ 0 < (𝑦 + 1))) | |
9 | breq2 5152 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
10 | id 22 | . . . . . 6 ⊢ (0 < 1 → 0 < 1) | |
11 | nnre 12271 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
12 | 11 | ad2antlr 727 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ) |
13 | 1red 11260 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 1 ∈ ℝ) | |
14 | simpr 484 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < 𝑦) | |
15 | simpll 767 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < 1) | |
16 | 12, 13, 14, 15 | sn-addgt0d 42454 | . . . . . 6 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < (𝑦 + 1)) |
17 | 6, 7, 8, 9, 10, 16 | nnindd 12284 | . . . . 5 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 0 < 𝐴) |
18 | 17 | gt0ne0d 11825 | . . . 4 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
19 | 18 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 0 < 1) → 𝐴 ≠ 0) |
20 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = 1 → (𝑥 < 0 ↔ 1 < 0)) | |
21 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 < 0 ↔ 𝑦 < 0)) | |
22 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 < 0 ↔ (𝑦 + 1) < 0)) | |
23 | breq1 5151 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 < 0 ↔ 𝐴 < 0)) | |
24 | id 22 | . . . . . 6 ⊢ (1 < 0 → 1 < 0) | |
25 | 11 | ad2antlr 727 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 𝑦 ∈ ℝ) |
26 | 1red 11260 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 1 ∈ ℝ) | |
27 | simpr 484 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 𝑦 < 0) | |
28 | simpll 767 | . . . . . . 7 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → 1 < 0) | |
29 | 25, 26, 27, 28 | sn-addlt0d 42453 | . . . . . 6 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → (𝑦 + 1) < 0) |
30 | 20, 21, 22, 23, 24, 29 | nnindd 12284 | . . . . 5 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 < 0) |
31 | 30 | lt0ne0d 11826 | . . . 4 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
32 | 31 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 1 < 0) → 𝐴 ≠ 0) |
33 | 19, 32 | jaodan 959 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ (0 < 1 ∨ 1 < 0)) → 𝐴 ≠ 0) |
34 | 5, 33 | mpan2 691 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ℕcn 12264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-nn 12265 df-2 12327 df-3 12328 df-resub 42373 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |