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Mirrors > Home > MPE Home > Th. List > infxrcl | Structured version Visualization version GIF version |
Description: The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
infxrcl | ⊢ (𝐴 ⊆ ℝ* → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12524 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
3 | xrinfmss 12693 | . 2 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
4 | 2, 3 | infcl 8941 | 1 ⊢ (𝐴 ⊆ ℝ* → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3935 Or wor 5467 infcinf 8894 ℝ*cxr 10663 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: infxrlb 12717 infxrgelb 12718 infxrre 12719 infxrmnf 12720 infxrss 12722 ixxlb 12750 limsupcl 14820 limsupval2 14827 imasdsf1olem 22912 nmoffn 23249 nmofval 23252 nmolb 23255 nmof 23257 metdsf 23385 ovolcl 24008 infrpge 41499 infxrbnd2 41517 infleinflem1 41518 infleinf 41520 infxrcld 41541 infxrpnf 41601 inficc 41690 liminfgord 41915 liminfgf 41919 liminfcl 41924 liminfval2 41929 liminflelimsuplem 41936 liminfvalxr 41944 ovnsubaddlem1 42733 ovolval5lem3 42817 |
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