Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 3halfnz | 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 9048 | . 2 ⊢ 1 ∈ ℤ | |
2 | 2cn 8759 | . . . . 5 ⊢ 2 ∈ ℂ | |
3 | 2 | mulid2i 7737 | . . . 4 ⊢ (1 · 2) = 2 |
4 | 2lt3 8858 | . . . 4 ⊢ 2 < 3 | |
5 | 3, 4 | eqbrtri 3919 | . . 3 ⊢ (1 · 2) < 3 |
6 | 1re 7733 | . . . 4 ⊢ 1 ∈ ℝ | |
7 | 3re 8762 | . . . 4 ⊢ 3 ∈ ℝ | |
8 | 2re 8758 | . . . . 5 ⊢ 2 ∈ ℝ | |
9 | 2pos 8779 | . . . . 5 ⊢ 0 < 2 | |
10 | 8, 9 | pm3.2i 270 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
11 | ltmuldiv 8600 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 · 2) < 3 ↔ 1 < (3 / 2))) | |
12 | 6, 7, 10, 11 | mp3an 1300 | . . 3 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
13 | 5, 12 | mpbi 144 | . 2 ⊢ 1 < (3 / 2) |
14 | 3lt4 8860 | . . . 4 ⊢ 3 < 4 | |
15 | 2t2e4 8842 | . . . . 5 ⊢ (2 · 2) = 4 | |
16 | 15 | breq2i 3907 | . . . 4 ⊢ (3 < (2 · 2) ↔ 3 < 4) |
17 | 14, 16 | mpbir 145 | . . 3 ⊢ 3 < (2 · 2) |
18 | 1p1e2 8805 | . . . . 5 ⊢ (1 + 1) = 2 | |
19 | 18 | breq2i 3907 | . . . 4 ⊢ ((3 / 2) < (1 + 1) ↔ (3 / 2) < 2) |
20 | ltdivmul 8602 | . . . . 5 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
21 | 7, 8, 10, 20 | mp3an 1300 | . . . 4 ⊢ ((3 / 2) < 2 ↔ 3 < (2 · 2)) |
22 | 19, 21 | bitri 183 | . . 3 ⊢ ((3 / 2) < (1 + 1) ↔ 3 < (2 · 2)) |
23 | 17, 22 | mpbir 145 | . 2 ⊢ (3 / 2) < (1 + 1) |
24 | btwnnz 9113 | . 2 ⊢ ((1 ∈ ℤ ∧ 1 < (3 / 2) ∧ (3 / 2) < (1 + 1)) → ¬ (3 / 2) ∈ ℤ) | |
25 | 1, 13, 23, 24 | mp3an 1300 | 1 ⊢ ¬ (3 / 2) ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 · cmul 7593 < clt 7768 / cdiv 8400 2c2 8739 3c3 8740 4c4 8741 ℤcz 9022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 |
This theorem is referenced by: nn0o1gt2 11529 |
Copyright terms: Public domain | W3C validator |