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 12731 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
3 | xrsupss 12899 | . 2 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
4 | 2, 3 | supcl 9074 | 1 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 Or wor 5467 supcsup 9056 ℝ*cxr 10866 < clt 10867 |
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-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: supxrun 12906 supxrmnf 12907 supxrbnd1 12911 supxrbnd2 12912 supxrub 12914 supxrleub 12916 supxrre 12917 supxrbnd 12918 supxrgtmnf 12919 supxrre1 12920 supxrre2 12921 supxrss 12922 ixxub 12956 limsupgord 15033 limsupcl 15034 limsupgf 15036 prdsdsf 23265 xpsdsval 23279 xrge0tsms 23731 elovolm 24372 ovolmge0 24374 ovolgelb 24377 ovollb2lem 24385 ovolunlem1a 24393 ovoliunlem1 24399 ovoliunlem2 24400 ovoliun 24402 ovolscalem1 24410 ovolicc1 24413 ovolicc2lem4 24417 voliunlem2 24448 voliunlem3 24449 ioombl1lem2 24456 uniioovol 24476 uniiccvol 24477 uniioombllem1 24478 uniioombllem3 24482 itg2cl 24630 itg2seq 24640 itg2monolem2 24649 itg2monolem3 24650 itg2mono 24651 mdeglt 24963 mdegxrcl 24965 radcnvcl 25309 nmoxr 28847 nmopxr 29947 nmfnxr 29960 xrofsup 30810 supxrnemnf 30811 xrge0tsmsd 31036 mblfinlem3 35553 mblfinlem4 35554 ismblfin 35555 itg2addnclem 35565 itg2gt0cn 35569 binomcxplemdvbinom 41644 binomcxplemcvg 41645 binomcxplemnotnn0 41647 supxrcld 42330 supxrgere 42545 supxrgelem 42549 supxrge 42550 suplesup 42551 suplesup2 42588 supxrcli 42647 liminfval2 42984 liminflelimsuplem 42991 sge0cl 43594 sge0xaddlem1 43646 sge0xaddlem2 43647 sge0reuz 43660 |
Copyright terms: Public domain | W3C validator |