Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzubioo | Structured version Visualization version GIF version |
Description: The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
uzubioo.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzubioo.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
uzubioo.3 | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
Ref | Expression |
---|---|
uzubioo | ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzubioo.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
2 | 1 | rexrd 10089 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
3 | pnfxr 10092 | . . . 4 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | 1 | ceilcld 39679 | . . . . . 6 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℤ) |
6 | 1zzd 11408 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
7 | 5, 6 | zaddcld 11486 | . . . . 5 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℤ) |
8 | 7 | zred 11482 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℝ) |
9 | uzubioo.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | 9 | zred 11482 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
11 | 8, 10 | ifcld 4131 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℝ) |
12 | 5 | zred 11482 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℝ) |
13 | 1 | ceilged 39673 | . . . . 5 ⊢ (𝜑 → 𝑋 ≤ (⌈‘𝑋)) |
14 | 12 | ltp1d 10954 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) < ((⌈‘𝑋) + 1)) |
15 | 1, 12, 8, 13, 14 | lelttrd 10195 | . . . 4 ⊢ (𝜑 → 𝑋 < ((⌈‘𝑋) + 1)) |
16 | 10, 8 | max2d 39688 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
17 | 1, 8, 11, 15, 16 | ltletrd 10197 | . . 3 ⊢ (𝜑 → 𝑋 < if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
18 | 11 | ltpnfd 11955 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) < +∞) |
19 | 2, 4, 11, 17, 18 | eliood 39720 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞)) |
20 | uzubioo.2 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
21 | 7, 9 | ifcld 4131 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℤ) |
22 | max1 12016 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ((⌈‘𝑋) + 1) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | |
23 | 10, 8, 22 | syl2anc 693 | . . 3 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) |
24 | 20, 9, 21, 23 | eluzd 39635 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) |
25 | eleq1 2689 | . . 3 ⊢ (𝑘 = if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) → (𝑘 ∈ 𝑍 ↔ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍)) | |
26 | 25 | rspcev 3309 | . 2 ⊢ ((if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞) ∧ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
27 | 19, 24, 26 | syl2anc 693 | 1 ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 ℤcz 11377 ℤ≥cuz 11687 (,)cioo 12175 ⌈cceil 12592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-ioo 12179 df-fl 12593 df-ceil 12594 |
This theorem is referenced by: uzubico 39795 uzubioo2 39796 |
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