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Mirrors > Home > MPE Home > Th. List > infssuzcl | Structured version Visualization version GIF version |
Description: The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
infssuzcl | ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzssz 11913 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
2 | zssre 11591 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3761 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℝ |
4 | sstr 3760 | . . . 4 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
5 | 3, 4 | mpan2 671 | . . 3 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ ℝ) |
6 | 5 | adantr 466 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ℝ) |
7 | uzwo 11959 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | |
8 | lbinfcl 11183 | . 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) → inf(𝑆, ℝ, < ) ∈ 𝑆) | |
9 | 6, 7, 8 | syl2anc 573 | 1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 ∃wrex 3062 ⊆ wss 3723 ∅c0 4063 class class class wbr 4787 ‘cfv 6030 infcinf 8507 ℝcr 10141 < clt 10280 ≤ cle 10281 ℤcz 11584 ℤ≥cuz 11893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-n0 11500 df-z 11585 df-uz 11894 |
This theorem is referenced by: zsupss 11985 uzwo3 11991 divalglem2 15326 bitsfzolem 15364 bezoutlem2 15465 lcmcllem 15517 lcmfval 15542 lcmfcllem 15546 odzcllem 15704 4sqlem13 15868 4sqlem14 15869 4sqlem17 15872 4sqlem18 15873 vdwnnlem3 15908 ramcl2lem 15920 ramtcl 15921 odlem1 18161 odlem2 18165 gexlem1 18201 gexlem2 18204 zringlpirlem2 20048 zringlpirlem3 20049 ovolicc2lem4 23508 iundisj 23536 ig1peu 24151 ig1pdvds 24156 elqaalem1 24294 elqaalem3 24296 ftalem4 25023 ftalem5 25024 iundisjf 29740 iundisjfi 29895 dgraalem 38239 allbutfiinf 40158 ioodvbdlimc1lem1 40659 fourierdlem31 40867 elaa2lem 40962 etransclem48 41011 |
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