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Mirrors > Home > MPE Home > Th. List > btwnnz | Structured version Visualization version GIF version |
Description: A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.) |
Ref | Expression |
---|---|
btwnnz | ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ∧ 𝐵 < (𝐴 + 1)) → ¬ 𝐵 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zltp1le 11843 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
2 | peano2z 11834 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈ ℤ) | |
3 | zre 11795 | . . . . . . . 8 ⊢ ((𝐴 + 1) ∈ ℤ → (𝐴 + 1) ∈ ℝ) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈ ℝ) |
5 | zre 11795 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
6 | lenlt 10517 | . . . . . . 7 ⊢ (((𝐴 + 1) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 1) ≤ 𝐵 ↔ ¬ 𝐵 < (𝐴 + 1))) | |
7 | 4, 5, 6 | syl2an 587 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 1) ≤ 𝐵 ↔ ¬ 𝐵 < (𝐴 + 1))) |
8 | 1, 7 | bitrd 271 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 < (𝐴 + 1))) |
9 | 8 | biimpd 221 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → ¬ 𝐵 < (𝐴 + 1))) |
10 | 9 | impancom 444 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 ∈ ℤ → ¬ 𝐵 < (𝐴 + 1))) |
11 | 10 | con2d 132 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 < (𝐴 + 1) → ¬ 𝐵 ∈ ℤ)) |
12 | 11 | 3impia 1098 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ∧ 𝐵 < (𝐴 + 1)) → ¬ 𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 class class class wbr 4925 (class class class)co 6974 ℝcr 10332 1c1 10334 + caddc 10336 < clt 10472 ≤ cle 10473 ℤcz 11791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 |
This theorem is referenced by: gtndiv 11870 3halfnz 11872 nn01to3 12153 seqcoll 13633 eirrlem 15415 nonsq 15953 zcld 23139 cosne0 24830 sumnnodd 41376 sinaover2ne0 41613 dirkertrigeqlem3 41850 dirkercncflem1 41853 dirkercncflem2 41854 dirkercncflem4 41856 fourierdlem20 41877 fourierdlem24 41881 fourierdlem46 41902 fourierdlem63 41919 fourierdlem64 41920 fourierdlem65 41921 |
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