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Mirrors > Home > MPE Home > Th. List > 3halfnz | Structured version Visualization version GIF version |
Description: Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) |
Ref | Expression |
---|---|
3halfnz | ⊢ ¬ (3 / 2) ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 11734 | . 2 ⊢ 1 ∈ ℤ | |
2 | 2cn 11425 | . . . . 5 ⊢ 2 ∈ ℂ | |
3 | 2 | mulid2i 10361 | . . . 4 ⊢ (1 · 2) = 2 |
4 | 2lt3 11529 | . . . 4 ⊢ 2 < 3 | |
5 | 3, 4 | eqbrtri 4893 | . . 3 ⊢ (1 · 2) < 3 |
6 | 1re 10355 | . . . 4 ⊢ 1 ∈ ℝ | |
7 | 3re 11430 | . . . 4 ⊢ 3 ∈ ℝ | |
8 | 2re 11424 | . . . . 5 ⊢ 2 ∈ ℝ | |
9 | 2pos 11460 | . . . . 5 ⊢ 0 < 2 | |
10 | 8, 9 | pm3.2i 464 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
11 | ltmuldiv 11225 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 · 2) < 3 ↔ 1 < (3 / 2))) | |
12 | 6, 7, 10, 11 | mp3an 1591 | . . 3 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
13 | 5, 12 | mpbi 222 | . 2 ⊢ 1 < (3 / 2) |
14 | 3lt4 11531 | . . . 4 ⊢ 3 < 4 | |
15 | 2t2e4 11521 | . . . . 5 ⊢ (2 · 2) = 4 | |
16 | 15 | breq2i 4880 | . . . 4 ⊢ (3 < (2 · 2) ↔ 3 < 4) |
17 | 14, 16 | mpbir 223 | . . 3 ⊢ 3 < (2 · 2) |
18 | 1p1e2 11482 | . . . . 5 ⊢ (1 + 1) = 2 | |
19 | 18 | breq2i 4880 | . . . 4 ⊢ ((3 / 2) < (1 + 1) ↔ (3 / 2) < 2) |
20 | ltdivmul 11227 | . . . . 5 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
21 | 7, 8, 10, 20 | mp3an 1591 | . . . 4 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
22 | 19, 21 | bitri 267 | . . 3 ⊢ ((3 / 2) < (1 + 1) ↔ 3 < (2 · 2)) |
23 | 17, 22 | mpbir 223 | . 2 ⊢ (3 / 2) < (1 + 1) |
24 | btwnnz 11780 | . 2 ⊢ ((1 ∈ ℤ ∧ 1 < (3 / 2) ∧ (3 / 2) < (1 + 1)) → ¬ (3 / 2) ∈ ℤ) | |
25 | 1, 13, 23, 24 | mp3an 1591 | 1 ⊢ ¬ (3 / 2) ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 class class class wbr 4872 (class class class)co 6904 ℝcr 10250 0cc0 10251 1c1 10252 + caddc 10254 · cmul 10256 < clt 10390 / cdiv 11008 2c2 11405 3c3 11406 4c4 11407 ℤcz 11703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-n0 11618 df-z 11704 |
This theorem is referenced by: nn0o1gt2 15470 |
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