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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 12639 | . 2 ⊢ 1 ∈ ℤ | |
2 | 2cn 12334 | . . . . 5 ⊢ 2 ∈ ℂ | |
3 | 2 | mullidi 11265 | . . . 4 ⊢ (1 · 2) = 2 |
4 | 2lt3 12431 | . . . 4 ⊢ 2 < 3 | |
5 | 3, 4 | eqbrtri 5173 | . . 3 ⊢ (1 · 2) < 3 |
6 | 1re 11260 | . . . 4 ⊢ 1 ∈ ℝ | |
7 | 3re 12339 | . . . 4 ⊢ 3 ∈ ℝ | |
8 | 2re 12333 | . . . . 5 ⊢ 2 ∈ ℝ | |
9 | 2pos 12362 | . . . . 5 ⊢ 0 < 2 | |
10 | 8, 9 | pm3.2i 469 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
11 | ltmuldiv 12134 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 · 2) < 3 ↔ 1 < (3 / 2))) | |
12 | 6, 7, 10, 11 | mp3an 1457 | . . 3 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
13 | 5, 12 | mpbi 229 | . 2 ⊢ 1 < (3 / 2) |
14 | 3lt4 12433 | . . . 4 ⊢ 3 < 4 | |
15 | 2t2e4 12423 | . . . . 5 ⊢ (2 · 2) = 4 | |
16 | 15 | breq2i 5160 | . . . 4 ⊢ (3 < (2 · 2) ↔ 3 < 4) |
17 | 14, 16 | mpbir 230 | . . 3 ⊢ 3 < (2 · 2) |
18 | 1p1e2 12384 | . . . . 5 ⊢ (1 + 1) = 2 | |
19 | 18 | breq2i 5160 | . . . 4 ⊢ ((3 / 2) < (1 + 1) ↔ (3 / 2) < 2) |
20 | ltdivmul 12136 | . . . . 5 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
21 | 7, 8, 10, 20 | mp3an 1457 | . . . 4 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
22 | 19, 21 | bitri 274 | . . 3 ⊢ ((3 / 2) < (1 + 1) ↔ 3 < (2 · 2)) |
23 | 17, 22 | mpbir 230 | . 2 ⊢ (3 / 2) < (1 + 1) |
24 | btwnnz 12685 | . 2 ⊢ ((1 ∈ ℤ ∧ 1 < (3 / 2) ∧ (3 / 2) < (1 + 1)) → ¬ (3 / 2) ∈ ℤ) | |
25 | 1, 13, 23, 24 | mp3an 1457 | 1 ⊢ ¬ (3 / 2) ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7423 ℝcr 11153 0cc0 11154 1c1 11155 + caddc 11157 · cmul 11159 < clt 11294 / cdiv 11917 2c2 12314 3c3 12315 4c4 12316 ℤcz 12605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-n0 12520 df-z 12606 |
This theorem is referenced by: n2dvds3 16368 nn0o1gt2 16378 |
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