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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 11950 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
2 | zssre 11673 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3807 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℝ |
4 | sstr 3806 | . . . 4 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
5 | 3, 4 | mpan2 683 | . . 3 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ ℝ) |
6 | 5 | adantr 473 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ℝ) |
7 | uzwo 11996 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | |
8 | lbinfcl 11269 | . 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) → inf(𝑆, ℝ, < ) ∈ 𝑆) | |
9 | 6, 7, 8 | syl2anc 580 | 1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 ⊆ wss 3769 ∅c0 4115 class class class wbr 4843 ‘cfv 6101 infcinf 8589 ℝcr 10223 < clt 10363 ≤ cle 10364 ℤcz 11666 ℤ≥cuz 11930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 |
This theorem is referenced by: zsupss 12022 uzwo3 12028 divalglem2 15454 bitsfzolem 15491 bezoutlem2 15592 lcmcllem 15644 lcmfval 15669 lcmfcllem 15673 odzcllem 15830 4sqlem13 15994 4sqlem14 15995 4sqlem17 15998 4sqlem18 15999 vdwnnlem3 16034 ramcl2lem 16046 ramtcl 16047 odlem1 18267 odlem2 18271 gexlem1 18307 gexlem2 18310 zringlpirlem2 20155 zringlpirlem3 20156 ovolicc2lem4 23628 iundisj 23656 ig1peu 24272 ig1pdvds 24277 elqaalem1 24415 elqaalem3 24417 ftalem4 25154 ftalem5 25155 iundisjf 29919 iundisjfi 30073 dgraalem 38500 allbutfiinf 40390 ioodvbdlimc1lem1 40890 fourierdlem31 41098 elaa2lem 41193 etransclem48 41242 |
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