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Mirrors > Home > MPE Home > Th. List > infxrlb | Structured version Visualization version GIF version |
Description: A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.) |
Ref | Expression |
---|---|
infxrlb | ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrcl 12923 | . . 3 ⊢ (𝐴 ⊆ ℝ* → inf(𝐴, ℝ*, < ) ∈ ℝ*) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
3 | ssel2 3895 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
4 | xrltso 12731 | . . . . 5 ⊢ < Or ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
6 | xrinfmss 12900 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
7 | 5, 6 | inflb 9105 | . . 3 ⊢ (𝐴 ⊆ ℝ* → (𝐵 ∈ 𝐴 → ¬ 𝐵 < inf(𝐴, ℝ*, < ))) |
8 | 7 | imp 410 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 < inf(𝐴, ℝ*, < )) |
9 | 2, 3, 8 | xrnltled 10901 | 1 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2110 ⊆ wss 3866 class class class wbr 5053 Or wor 5467 infcinf 9057 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 |
This theorem is referenced by: infxrre 12926 infxrmnf 12927 infxrss 12929 ixxlb 12957 limsupval2 15041 imasdsf1olem 23271 ovollb 24376 ovolsslem 24381 infleinflem1 42582 infxrlbrnmpt2 42623 infleinf2 42627 infxrlesupxr 42649 inficc 42747 ressiooinf 42770 liminfgord 42970 cnrefiisplem 43045 ioorrnopnlem 43520 ovnlecvr 43771 ovn0lem 43778 ovnhoilem1 43814 ovnlecvr2 43823 |
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