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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 12873 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
2 | zssre 12595 | . . . 4 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3982 | . . 3 ⊢ (ℤ≥‘𝑀) ⊆ ℝ |
4 | sstr 3981 | . . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
5 | 3, 4 | mpan2 689 | . 2 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ ℝ) |
6 | uzwo 12925 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | |
7 | lbinfcl 12198 | . 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) → inf(𝑆, ℝ, < ) ∈ 𝑆) | |
8 | 5, 6, 7 | syl2an2r 683 | 1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ⊆ wss 3939 ∅c0 4318 class class class wbr 5143 ‘cfv 6543 infcinf 9464 ℝcr 11137 < clt 11278 ≤ cle 11279 ℤcz 12588 ℤ≥cuz 12852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 |
This theorem is referenced by: zsupss 12951 uzwo3 12957 divalglem2 16371 bitsfzolem 16408 bezoutlem2 16515 lcmcllem 16566 lcmfval 16591 lcmfcllem 16595 odzcllem 16760 4sqlem13 16925 4sqlem14 16926 4sqlem17 16929 4sqlem18 16930 vdwnnlem3 16965 ramcl2lem 16977 ramtcl 16978 odfval 19491 odlem1 19494 odlem2 19498 gexlem1 19538 gexlem2 19541 zringlpirlem2 21393 zringlpirlem3 21394 ovolicc2lem4 25467 iundisj 25495 ig1peu 26127 ig1pdvds 26132 elqaalem1 26272 elqaalem3 26274 ftalem4 27026 ftalem5 27027 iundisjf 32424 iundisjfi 32609 dgraalem 42634 allbutfiinf 44865 ioodvbdlimc1lem1 45382 fourierdlem31 45589 elaa2lem 45684 etransclem48 45733 |
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