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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 11312 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ*) | 
| 3 | pnfxr 11316 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) | 
| 5 | 1 | ceilcld 13884 | . . . . . 6 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℤ) | 
| 6 | 1zzd 12650 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 7 | 5, 6 | zaddcld 12728 | . . . . 5 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℤ) | 
| 8 | 7 | zred 12724 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ∈ ℝ) | 
| 9 | uzubioo.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 10 | 9 | zred 12724 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | 
| 11 | 8, 10 | ifcld 4571 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℝ) | 
| 12 | 5 | zred 12724 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) ∈ ℝ) | 
| 13 | 1 | ceilged 13887 | . . . . 5 ⊢ (𝜑 → 𝑋 ≤ (⌈‘𝑋)) | 
| 14 | 12 | ltp1d 12199 | . . . . 5 ⊢ (𝜑 → (⌈‘𝑋) < ((⌈‘𝑋) + 1)) | 
| 15 | 1, 12, 8, 13, 14 | lelttrd 11420 | . . . 4 ⊢ (𝜑 → 𝑋 < ((⌈‘𝑋) + 1)) | 
| 16 | 10, 8 | max2d 45474 | . . . 4 ⊢ (𝜑 → ((⌈‘𝑋) + 1) ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | 
| 17 | 1, 8, 11, 15, 16 | ltletrd 11422 | . . 3 ⊢ (𝜑 → 𝑋 < if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | 
| 18 | 11 | ltpnfd 13164 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) < +∞) | 
| 19 | 2, 4, 11, 17, 18 | eliood 45516 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ (𝑋(,)+∞)) | 
| 20 | uzubioo.2 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 21 | 7, 9 | ifcld 4571 | . . 3 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ ℤ) | 
| 22 | max1 13228 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ((⌈‘𝑋) + 1) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | |
| 23 | 10, 8, 22 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀)) | 
| 24 | 20, 9, 21, 23 | eluzd 45425 | . 2 ⊢ (𝜑 → if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍) | 
| 25 | eleq1 2828 | . . 3 ⊢ (𝑘 = if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) → (𝑘 ∈ 𝑍 ↔ if(𝑀 ≤ ((⌈‘𝑋) + 1), ((⌈‘𝑋) + 1), 𝑀) ∈ 𝑍)) | |
| 26 | 25 | rspcev 3621 | . 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 584 | 1 ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ifcif 4524 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 1c1 11157 + caddc 11159 +∞cpnf 11293 ℝ*cxr 11295 ≤ cle 11297 ℤcz 12615 ℤ≥cuz 12879 (,)cioo 13388 ⌈cceil 13832 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-ioo 13392 df-fl 13833 df-ceil 13834 | 
| This theorem is referenced by: uzubico 45586 uzubioo2 45587 | 
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