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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 8978 | . 2 ⊢ 1 ∈ ℤ | |
2 | 2cn 8695 | . . . . 5 ⊢ 2 ∈ ℂ | |
3 | 2 | mulid2i 7687 | . . . 4 ⊢ (1 · 2) = 2 |
4 | 2lt3 8788 | . . . 4 ⊢ 2 < 3 | |
5 | 3, 4 | eqbrtri 3912 | . . 3 ⊢ (1 · 2) < 3 |
6 | 1re 7683 | . . . 4 ⊢ 1 ∈ ℝ | |
7 | 3re 8698 | . . . 4 ⊢ 3 ∈ ℝ | |
8 | 2re 8694 | . . . . 5 ⊢ 2 ∈ ℝ | |
9 | 2pos 8715 | . . . . 5 ⊢ 0 < 2 | |
10 | 8, 9 | pm3.2i 268 | . . . 4 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
11 | ltmuldiv 8536 | . . . 4 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 · 2) < 3 ↔ 1 < (3 / 2))) | |
12 | 6, 7, 10, 11 | mp3an 1296 | . . 3 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
13 | 5, 12 | mpbi 144 | . 2 ⊢ 1 < (3 / 2) |
14 | 3lt4 8790 | . . . 4 ⊢ 3 < 4 | |
15 | 2t2e4 8772 | . . . . 5 ⊢ (2 · 2) = 4 | |
16 | 15 | breq2i 3901 | . . . 4 ⊢ (3 < (2 · 2) ↔ 3 < 4) |
17 | 14, 16 | mpbir 145 | . . 3 ⊢ 3 < (2 · 2) |
18 | 1p1e2 8741 | . . . . 5 ⊢ (1 + 1) = 2 | |
19 | 18 | breq2i 3901 | . . . 4 ⊢ ((3 / 2) < (1 + 1) ↔ (3 / 2) < 2) |
20 | ltdivmul 8538 | . . . . 5 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((3 / 2) < 2 ↔ 3 < (2 · 2))) | |
21 | 7, 8, 10, 20 | mp3an 1296 | . . . 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 9043 | . 2 ⊢ ((1 ∈ ℤ ∧ 1 < (3 / 2) ∧ (3 / 2) < (1 + 1)) → ¬ (3 / 2) ∈ ℤ) | |
25 | 1, 13, 23, 24 | mp3an 1296 | 1 ⊢ ¬ (3 / 2) ∈ ℤ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∈ wcel 1461 class class class wbr 3893 (class class class)co 5726 ℝcr 7540 0cc0 7541 1c1 7542 + caddc 7544 · cmul 7546 < clt 7718 / cdiv 8339 2c2 8675 3c3 8676 4c4 8677 ℤcz 8952 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-id 4173 df-po 4176 df-iso 4177 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-n0 8876 df-z 8953 |
This theorem is referenced by: nn0o1gt2 11444 |
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