![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13107 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
3 | xrsupss 13275 | . 2 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
4 | 2, 3 | supcl 9440 | 1 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3946 Or wor 5583 supcsup 9422 ℝ*cxr 11234 < clt 11235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 |
This theorem is referenced by: supxrun 13282 supxrmnf 13283 supxrbnd1 13287 supxrbnd2 13288 supxrub 13290 supxrleub 13292 supxrre 13293 supxrbnd 13294 supxrgtmnf 13295 supxrre1 13296 supxrre2 13297 supxrss 13298 ixxub 13332 limsupgord 15403 limsupcl 15404 limsupgf 15406 prdsdsf 23842 xpsdsval 23856 xrge0tsms 24319 elovolm 24961 ovolmge0 24963 ovolgelb 24966 ovollb2lem 24974 ovolunlem1a 24982 ovoliunlem1 24988 ovoliunlem2 24989 ovoliun 24991 ovolscalem1 24999 ovolicc1 25002 ovolicc2lem4 25006 voliunlem2 25037 voliunlem3 25038 ioombl1lem2 25045 uniioovol 25065 uniiccvol 25066 uniioombllem1 25067 uniioombllem3 25071 itg2cl 25219 itg2seq 25229 itg2monolem2 25238 itg2monolem3 25239 itg2mono 25240 mdeglt 25552 mdegxrcl 25554 radcnvcl 25898 nmoxr 29984 nmopxr 31084 nmfnxr 31097 xrofsup 31951 supxrnemnf 31952 xrge0tsmsd 32180 mblfinlem3 36432 mblfinlem4 36433 ismblfin 36434 itg2addnclem 36444 itg2gt0cn 36448 binomcxplemdvbinom 42983 binomcxplemcvg 42984 binomcxplemnotnn0 42986 supxrcld 43667 supxrgere 43916 supxrgelem 43920 supxrge 43921 suplesup 43922 suplesup2 43959 supxrcli 44017 liminfval2 44357 liminflelimsuplem 44364 sge0cl 44970 sge0xaddlem1 45022 sge0xaddlem2 45023 sge0reuz 45036 |
Copyright terms: Public domain | W3C validator |