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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 12252 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
2 | zssre 11976 | . . . 4 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3924 | . . 3 ⊢ (ℤ≥‘𝑀) ⊆ ℝ |
4 | sstr 3923 | . . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
5 | 3, 4 | mpan2 690 | . 2 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ ℝ) |
6 | uzwo 12299 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | |
7 | lbinfcl 11582 | . 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) → inf(𝑆, ℝ, < ) ∈ 𝑆) | |
8 | 5, 6, 7 | syl2an2r 684 | 1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 infcinf 8889 ℝcr 10525 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: zsupss 12325 uzwo3 12331 divalglem2 15736 bitsfzolem 15773 bezoutlem2 15878 lcmcllem 15930 lcmfval 15955 lcmfcllem 15959 odzcllem 16119 4sqlem13 16283 4sqlem14 16284 4sqlem17 16287 4sqlem18 16288 vdwnnlem3 16323 ramcl2lem 16335 ramtcl 16336 odfval 18652 odlem1 18655 odlem2 18659 gexlem1 18696 gexlem2 18699 zringlpirlem2 20178 zringlpirlem3 20179 ovolicc2lem4 24124 iundisj 24152 ig1peu 24772 ig1pdvds 24777 elqaalem1 24915 elqaalem3 24917 ftalem4 25661 ftalem5 25662 iundisjf 30352 iundisjfi 30545 dgraalem 40089 allbutfiinf 42057 ioodvbdlimc1lem1 42573 fourierdlem31 42780 elaa2lem 42875 etransclem48 42924 |
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