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Mirrors > Home > MPE Home > Th. List > xrltso | Structured version Visualization version GIF version |
Description: 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
xrltso | ⊢ < Or ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlttri 11972 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥))) | |
2 | xrlttr 11973 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
3 | 1, 2 | isso2i 5067 | 1 ⊢ < Or ℝ* |
Colors of variables: wff setvar class |
Syntax hints: Or wor 5034 ℝ*cxr 10073 < clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 |
This theorem is referenced by: xrlttri2 11975 xrlttri3 11976 xrltne 11994 xmullem 12094 xmulasslem 12115 supxr 12143 supxrcl 12145 supxrun 12146 supxrmnf 12147 supxrunb1 12149 supxrunb2 12150 supxrub 12154 supxrlub 12155 infxrcl 12163 infxrlb 12164 infxrgelb 12165 xrinf0 12168 infmremnf 12173 limsupval 14205 limsupgval 14207 limsupgre 14212 ramval 15712 ramcl2lem 15713 prdsdsfn 16125 prdsdsval 16138 imasdsfn 16174 imasdsval 16175 prdsmet 22175 xpsdsval 22186 prdsbl 22296 tmsxpsval2 22344 nmoval 22519 xrge0tsms2 22638 metdsval 22650 iccpnfhmeo 22744 xrhmeo 22745 ovolval 23242 ovolf 23250 ovolctb 23258 itg2val 23495 mdegval 23823 mdegldg 23826 mdegxrf 23828 mdegcl 23829 aannenlem2 24084 nmooval 27618 nmoo0 27646 nmopval 28715 nmfnval 28735 nmop0 28845 nmfn0 28846 xrsupssd 29524 xrge0infssd 29526 infxrge0lb 29529 infxrge0glb 29530 infxrge0gelb 29531 xrsclat 29680 xrge0iifiso 29981 esumval 30108 esumnul 30110 esum0 30111 gsumesum 30121 esumsnf 30126 esumpcvgval 30140 esum2d 30155 omsfval 30356 omsf 30358 oms0 30359 omssubaddlem 30361 omssubadd 30362 mblfinlem2 33447 ovoliunnfl 33451 voliunnfl 33453 volsupnfl 33454 itg2addnclem 33461 radcnvrat 38513 infxrglb 39556 xrgtso 39561 infxr 39583 infxrunb2 39584 infxrpnf 39674 limsup0 39926 limsuppnfdlem 39933 limsupequzlem 39954 supcnvlimsup 39972 limsuplt2 39985 liminfval 39991 limsupge 39993 liminfgval 39994 liminfval2 40000 limsup10ex 40005 liminf10ex 40006 liminflelimsuplem 40007 cnrefiisplem 40055 etransclem48 40499 sge0val 40583 sge0z 40592 sge00 40593 sge0sn 40596 sge0tsms 40597 ovnval2 40759 smflimsuplem1 41026 smflimsuplem2 41027 smflimsuplem4 41029 smflimsuplem7 41032 |
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