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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-nnne0 | Structured version Visualization version GIF version | ||
| Description: nnne0 12300 without ax-mulcom 11219. (Contributed by SN, 25-Jan-2025.) |
| Ref | Expression |
|---|---|
| sn-nnne0 | ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ne1 12337 | . . 3 ⊢ 0 ≠ 1 | |
| 2 | 0re 11263 | . . . 4 ⊢ 0 ∈ ℝ | |
| 3 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 4 | 2, 3 | lttri2i 11375 | . . 3 ⊢ (0 ≠ 1 ↔ (0 < 1 ∨ 1 < 0)) |
| 5 | 1, 4 | mpbi 230 | . 2 ⊢ (0 < 1 ∨ 1 < 0) |
| 6 | breq2 5147 | . . . . . 6 ⊢ (𝑥 = 1 → (0 < 𝑥 ↔ 0 < 1)) | |
| 7 | breq2 5147 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (0 < 𝑥 ↔ 0 < 𝑦)) | |
| 8 | breq2 5147 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (0 < 𝑥 ↔ 0 < (𝑦 + 1))) | |
| 9 | breq2 5147 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
| 10 | id 22 | . . . . . 6 ⊢ (0 < 1 → 0 < 1) | |
| 11 | nnre 12273 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
| 12 | 11 | ad2antlr 727 | . . . . . . 7 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ) |
| 13 | 1red 11262 | . . . . . . 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 42477 | . . . . . 6 ⊢ (((0 < 1 ∧ 𝑦 ∈ ℕ) ∧ 0 < 𝑦) → 0 < (𝑦 + 1)) |
| 17 | 6, 7, 8, 9, 10, 16 | nnindd 12286 | . . . . 5 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 0 < 𝐴) |
| 18 | 17 | gt0ne0d 11827 | . . . 4 ⊢ ((0 < 1 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
| 19 | 18 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 0 < 1) → 𝐴 ≠ 0) |
| 20 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 1 → (𝑥 < 0 ↔ 1 < 0)) | |
| 21 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 < 0 ↔ 𝑦 < 0)) | |
| 22 | breq1 5146 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 < 0 ↔ (𝑦 + 1) < 0)) | |
| 23 | breq1 5146 | . . . . . 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 11262 | . . . . . . 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 42476 | . . . . . 6 ⊢ (((1 < 0 ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 0) → (𝑦 + 1) < 0) |
| 30 | 20, 21, 22, 23, 24, 29 | nnindd 12286 | . . . . 5 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 < 0) |
| 31 | 30 | lt0ne0d 11828 | . . . 4 ⊢ ((1 < 0 ∧ 𝐴 ∈ ℕ) → 𝐴 ≠ 0) |
| 32 | 31 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 1 < 0) → 𝐴 ≠ 0) |
| 33 | 19, 32 | jaodan 960 | . 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 848 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ℕcn 12266 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-resub 42396 |
| This theorem is referenced by: (None) |
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