![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-0ne2 | Structured version Visualization version GIF version |
Description: 0ne2 12441 without ax-mulcom 11194. (Contributed by SN, 23-Jan-2024.) |
Ref | Expression |
---|---|
sn-0ne2 | ⊢ 0 ≠ 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11236 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | readdlid 41880 | . . . 4 ⊢ (1 ∈ ℝ → (0 + 1) = 1) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (0 + 1) = 1 |
4 | sn-1ne2 41762 | . . . . . 6 ⊢ 1 ≠ 2 | |
5 | 2re 12308 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
6 | 1, 5 | lttri2i 11350 | . . . . . 6 ⊢ (1 ≠ 2 ↔ (1 < 2 ∨ 2 < 1)) |
7 | 4, 6 | mpbi 229 | . . . . 5 ⊢ (1 < 2 ∨ 2 < 1) |
8 | 1red 11237 | . . . . . . 7 ⊢ (1 < 2 → 1 ∈ ℝ) | |
9 | 1, 5, 1 | ltadd2i 11367 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ (1 + 1) < (1 + 2)) |
10 | 9 | biimpi 215 | . . . . . . . . 9 ⊢ (1 < 2 → (1 + 1) < (1 + 2)) |
11 | 1p1e2 12359 | . . . . . . . . 9 ⊢ (1 + 1) = 2 | |
12 | 1p2e3 12377 | . . . . . . . . 9 ⊢ (1 + 2) = 3 | |
13 | 10, 11, 12 | 3brtr3g 5175 | . . . . . . . 8 ⊢ (1 < 2 → 2 < 3) |
14 | 3re 12314 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
15 | 1, 5, 14 | lttri 11362 | . . . . . . . 8 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
16 | 13, 15 | mpdan 686 | . . . . . . 7 ⊢ (1 < 2 → 1 < 3) |
17 | 8, 16 | ltned 11372 | . . . . . 6 ⊢ (1 < 2 → 1 ≠ 3) |
18 | 14 | a1i 11 | . . . . . . 7 ⊢ (2 < 1 → 3 ∈ ℝ) |
19 | 5, 1, 1 | ltadd2i 11367 | . . . . . . . . . 10 ⊢ (2 < 1 ↔ (1 + 2) < (1 + 1)) |
20 | 19 | biimpi 215 | . . . . . . . . 9 ⊢ (2 < 1 → (1 + 2) < (1 + 1)) |
21 | 20, 12, 11 | 3brtr3g 5175 | . . . . . . . 8 ⊢ (2 < 1 → 3 < 2) |
22 | 14, 5, 1 | lttri 11362 | . . . . . . . 8 ⊢ ((3 < 2 ∧ 2 < 1) → 3 < 1) |
23 | 21, 22 | mpancom 687 | . . . . . . 7 ⊢ (2 < 1 → 3 < 1) |
24 | 18, 23 | gtned 11371 | . . . . . 6 ⊢ (2 < 1 → 1 ≠ 3) |
25 | 17, 24 | jaoi 856 | . . . . 5 ⊢ ((1 < 2 ∨ 2 < 1) → 1 ≠ 3) |
26 | 7, 25 | ax-mp 5 | . . . 4 ⊢ 1 ≠ 3 |
27 | df-3 12298 | . . . 4 ⊢ 3 = (2 + 1) | |
28 | 26, 27 | neeqtri 3008 | . . 3 ⊢ 1 ≠ (2 + 1) |
29 | 3, 28 | eqnetri 3006 | . 2 ⊢ (0 + 1) ≠ (2 + 1) |
30 | oveq1 7421 | . . 3 ⊢ (0 = 2 → (0 + 1) = (2 + 1)) | |
31 | 30 | necon3i 2968 | . 2 ⊢ ((0 + 1) ≠ (2 + 1) → 0 ≠ 2) |
32 | 29, 31 | ax-mp 5 | 1 ⊢ 0 ≠ 2 |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 (class class class)co 7414 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 2c2 12289 3c3 12290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-ltxr 11275 df-2 12297 df-3 12298 df-resub 41843 |
This theorem is referenced by: remul01 41884 sn-0tie0 41916 |
Copyright terms: Public domain | W3C validator |