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Mirrors > Home > MPE Home > Th. List > supxrcl | Structured version Visualization version GIF version |
Description: The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.) |
Ref | Expression |
---|---|
supxrcl | ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12535 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
3 | xrsupss 12703 | . 2 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
4 | 2, 3 | supcl 8922 | 1 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 Or wor 5473 supcsup 8904 ℝ*cxr 10674 < clt 10675 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 |
This theorem is referenced by: supxrun 12710 supxrmnf 12711 supxrbnd1 12715 supxrbnd2 12716 supxrub 12718 supxrleub 12720 supxrre 12721 supxrbnd 12722 supxrgtmnf 12723 supxrre1 12724 supxrre2 12725 supxrss 12726 ixxub 12760 limsupgord 14829 limsupcl 14830 limsupgf 14832 prdsdsf 22977 xpsdsval 22991 xrge0tsms 23442 elovolm 24076 ovolmge0 24078 ovolgelb 24081 ovollb2lem 24089 ovolunlem1a 24097 ovoliunlem1 24103 ovoliunlem2 24104 ovoliun 24106 ovolscalem1 24114 ovolicc1 24117 ovolicc2lem4 24121 voliunlem2 24152 voliunlem3 24153 ioombl1lem2 24160 uniioovol 24180 uniiccvol 24181 uniioombllem1 24182 uniioombllem3 24186 itg2cl 24333 itg2seq 24343 itg2monolem2 24352 itg2monolem3 24353 itg2mono 24354 mdeglt 24659 mdegxrcl 24661 radcnvcl 25005 nmoxr 28543 nmopxr 29643 nmfnxr 29656 xrofsup 30492 supxrnemnf 30493 xrge0tsmsd 30692 mblfinlem3 34946 mblfinlem4 34947 ismblfin 34948 itg2addnclem 34958 itg2gt0cn 34962 binomcxplemdvbinom 40705 binomcxplemcvg 40706 binomcxplemnotnn0 40708 supxrcld 41393 supxrgere 41621 supxrgelem 41625 supxrge 41626 suplesup 41627 suplesup2 41664 supxrcli 41728 liminfval2 42069 liminflelimsuplem 42076 sge0cl 42683 sge0xaddlem1 42735 sge0xaddlem2 42736 sge0reuz 42749 |
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