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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-0ne2 | Structured version Visualization version GIF version |
Description: 0ne2 11923 without ax-mulcom 10679. (Contributed by SN, 23-Jan-2024.) |
Ref | Expression |
---|---|
sn-0ne2 | ⊢ 0 ≠ 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10719 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | readdid2 39983 | . . . 4 ⊢ (1 ∈ ℝ → (0 + 1) = 1) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (0 + 1) = 1 |
4 | sn-1ne2 39891 | . . . . . 6 ⊢ 1 ≠ 2 | |
5 | 2re 11790 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
6 | 1, 5 | lttri2i 10832 | . . . . . 6 ⊢ (1 ≠ 2 ↔ (1 < 2 ∨ 2 < 1)) |
7 | 4, 6 | mpbi 233 | . . . . 5 ⊢ (1 < 2 ∨ 2 < 1) |
8 | 1red 10720 | . . . . . . 7 ⊢ (1 < 2 → 1 ∈ ℝ) | |
9 | 1, 5, 1 | ltadd2i 10849 | . . . . . . . . . 10 ⊢ (1 < 2 ↔ (1 + 1) < (1 + 2)) |
10 | 9 | biimpi 219 | . . . . . . . . 9 ⊢ (1 < 2 → (1 + 1) < (1 + 2)) |
11 | 1p1e2 11841 | . . . . . . . . 9 ⊢ (1 + 1) = 2 | |
12 | 1p2e3 11859 | . . . . . . . . 9 ⊢ (1 + 2) = 3 | |
13 | 10, 11, 12 | 3brtr3g 5063 | . . . . . . . 8 ⊢ (1 < 2 → 2 < 3) |
14 | 3re 11796 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
15 | 1, 5, 14 | lttri 10844 | . . . . . . . 8 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
16 | 13, 15 | mpdan 687 | . . . . . . 7 ⊢ (1 < 2 → 1 < 3) |
17 | 8, 16 | ltned 10854 | . . . . . 6 ⊢ (1 < 2 → 1 ≠ 3) |
18 | 14 | a1i 11 | . . . . . . 7 ⊢ (2 < 1 → 3 ∈ ℝ) |
19 | 5, 1, 1 | ltadd2i 10849 | . . . . . . . . . 10 ⊢ (2 < 1 ↔ (1 + 2) < (1 + 1)) |
20 | 19 | biimpi 219 | . . . . . . . . 9 ⊢ (2 < 1 → (1 + 2) < (1 + 1)) |
21 | 20, 12, 11 | 3brtr3g 5063 | . . . . . . . 8 ⊢ (2 < 1 → 3 < 2) |
22 | 14, 5, 1 | lttri 10844 | . . . . . . . 8 ⊢ ((3 < 2 ∧ 2 < 1) → 3 < 1) |
23 | 21, 22 | mpancom 688 | . . . . . . 7 ⊢ (2 < 1 → 3 < 1) |
24 | 18, 23 | gtned 10853 | . . . . . 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 11780 | . . . 4 ⊢ 3 = (2 + 1) | |
28 | 26, 27 | neeqtri 3006 | . . 3 ⊢ 1 ≠ (2 + 1) |
29 | 3, 28 | eqnetri 3004 | . 2 ⊢ (0 + 1) ≠ (2 + 1) |
30 | oveq1 7177 | . . 3 ⊢ (0 = 2 → (0 + 1) = (2 + 1)) | |
31 | 30 | necon3i 2966 | . 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 2114 ≠ wne 2934 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 < clt 10753 2c2 11771 3c3 11772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-ltxr 10758 df-2 11779 df-3 11780 df-resub 39946 |
This theorem is referenced by: remul01 39987 sn-0tie0 40018 |
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