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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-0ne2 | Structured version Visualization version GIF version |
Description: 0ne2 12367 without ax-mulcom 11122. (Contributed by SN, 23-Jan-2024.) |
Ref | Expression |
---|---|
sn-0ne2 | ⊢ 0 ≠ 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11162 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | readdid2 40901 | . . . 4 ⊢ (1 ∈ ℝ → (0 + 1) = 1) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (0 + 1) = 1 |
4 | sn-1ne2 40810 | . . . . . 6 ⊢ 1 ≠ 2 | |
5 | 2re 12234 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
6 | 1, 5 | lttri2i 11276 | . . . . . 6 ⊢ (1 ≠ 2 ↔ (1 < 2 ∨ 2 < 1)) |
7 | 4, 6 | mpbi 229 | . . . . 5 ⊢ (1 < 2 ∨ 2 < 1) |
8 | 1red 11163 | . . . . . . 7 ⊢ (1 < 2 → 1 ∈ ℝ) | |
9 | 1, 5, 1 | ltadd2i 11293 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ (1 + 1) < (1 + 2)) |
10 | 9 | biimpi 215 | . . . . . . . . 9 ⊢ (1 < 2 → (1 + 1) < (1 + 2)) |
11 | 1p1e2 12285 | . . . . . . . . 9 ⊢ (1 + 1) = 2 | |
12 | 1p2e3 12303 | . . . . . . . . 9 ⊢ (1 + 2) = 3 | |
13 | 10, 11, 12 | 3brtr3g 5143 | . . . . . . . 8 ⊢ (1 < 2 → 2 < 3) |
14 | 3re 12240 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
15 | 1, 5, 14 | lttri 11288 | . . . . . . . 8 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
16 | 13, 15 | mpdan 686 | . . . . . . 7 ⊢ (1 < 2 → 1 < 3) |
17 | 8, 16 | ltned 11298 | . . . . . 6 ⊢ (1 < 2 → 1 ≠ 3) |
18 | 14 | a1i 11 | . . . . . . 7 ⊢ (2 < 1 → 3 ∈ ℝ) |
19 | 5, 1, 1 | ltadd2i 11293 | . . . . . . . . . 10 ⊢ (2 < 1 ↔ (1 + 2) < (1 + 1)) |
20 | 19 | biimpi 215 | . . . . . . . . 9 ⊢ (2 < 1 → (1 + 2) < (1 + 1)) |
21 | 20, 12, 11 | 3brtr3g 5143 | . . . . . . . 8 ⊢ (2 < 1 → 3 < 2) |
22 | 14, 5, 1 | lttri 11288 | . . . . . . . 8 ⊢ ((3 < 2 ∧ 2 < 1) → 3 < 1) |
23 | 21, 22 | mpancom 687 | . . . . . . 7 ⊢ (2 < 1 → 3 < 1) |
24 | 18, 23 | gtned 11297 | . . . . . 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 12224 | . . . 4 ⊢ 3 = (2 + 1) | |
28 | 26, 27 | neeqtri 3017 | . . 3 ⊢ 1 ≠ (2 + 1) |
29 | 3, 28 | eqnetri 3015 | . 2 ⊢ (0 + 1) ≠ (2 + 1) |
30 | oveq1 7369 | . . 3 ⊢ (0 = 2 → (0 + 1) = (2 + 1)) | |
31 | 30 | necon3i 2977 | . 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 1542 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5110 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 2c2 12215 3c3 12216 |
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 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
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-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-2 12223 df-3 12224 df-resub 40864 |
This theorem is referenced by: remul01 40905 sn-0tie0 40937 |
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