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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 12850 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
2 | zssre 12572 | . . . 4 ⊢ ℤ ⊆ ℝ | |
3 | 1, 2 | sstri 3991 | . . 3 ⊢ (ℤ≥‘𝑀) ⊆ ℝ |
4 | sstr 3990 | . . 3 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℝ) → 𝑆 ⊆ ℝ) | |
5 | 3, 4 | mpan2 688 | . 2 ⊢ (𝑆 ⊆ (ℤ≥‘𝑀) → 𝑆 ⊆ ℝ) |
6 | uzwo 12902 | . 2 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | |
7 | lbinfcl 12175 | . 2 ⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) → inf(𝑆, ℝ, < ) ∈ 𝑆) | |
8 | 5, 6, 7 | syl2an2r 682 | 1 ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 infcinf 9442 ℝcr 11115 < clt 11255 ≤ cle 11256 ℤcz 12565 ℤ≥cuz 12829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 |
This theorem is referenced by: zsupss 12928 uzwo3 12934 divalglem2 16345 bitsfzolem 16382 bezoutlem2 16489 lcmcllem 16540 lcmfval 16565 lcmfcllem 16569 odzcllem 16732 4sqlem13 16897 4sqlem14 16898 4sqlem17 16901 4sqlem18 16902 vdwnnlem3 16937 ramcl2lem 16949 ramtcl 16950 odfval 19448 odlem1 19451 odlem2 19455 gexlem1 19495 gexlem2 19498 zringlpirlem2 21322 zringlpirlem3 21323 ovolicc2lem4 25368 iundisj 25396 ig1peu 26026 ig1pdvds 26031 elqaalem1 26170 elqaalem3 26172 ftalem4 26920 ftalem5 26921 iundisjf 32252 iundisjfi 32439 dgraalem 42349 allbutfiinf 44588 ioodvbdlimc1lem1 45105 fourierdlem31 45312 elaa2lem 45407 etransclem48 45456 |
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