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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 12873 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥))) | |
2 | xrlttr 12874 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
3 | 1, 2 | isso2i 5538 | 1 ⊢ < Or ℝ* |
Colors of variables: wff setvar class |
Syntax hints: Or wor 5502 ℝ*cxr 11008 < clt 11009 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 |
This theorem is referenced by: xrlttri2 12876 xrlttri3 12877 xrltne 12897 xmullem 12998 xmulasslem 13019 supxr 13047 supxrcl 13049 supxrun 13050 supxrmnf 13051 supxrunb1 13053 supxrunb2 13054 supxrub 13058 supxrlub 13059 infxrcl 13067 infxrlb 13068 infxrgelb 13069 xrinf0 13072 infmremnf 13077 limsupval 15183 limsupgval 15185 limsupgre 15190 ramval 16709 ramcl2lem 16710 prdsdsfn 17176 prdsdsval 17189 imasdsfn 17225 imasdsval 17226 prdsmet 23523 xpsdsval 23534 prdsbl 23647 tmsxpsval2 23695 nmoval 23879 xrge0tsms2 23998 metdsval 24010 iccpnfhmeo 24108 xrhmeo 24109 ovolval 24637 ovolf 24646 ovolctb 24654 itg2val 24893 mdegval 25228 mdegldg 25231 mdegxrf 25233 mdegcl 25234 aannenlem2 25489 nmooval 29125 nmoo0 29153 nmopval 30218 nmfnval 30238 nmop0 30348 nmfn0 30349 xrsupssd 31082 xrge0infssd 31084 infxrge0lb 31087 infxrge0glb 31088 infxrge0gelb 31089 xrsclat 31289 xrge0iifiso 31885 esumval 32014 esumnul 32016 esum0 32017 gsumesum 32027 esumsnf 32032 esumpcvgval 32046 esum2d 32061 omsfval 32261 omsf 32263 oms0 32264 omssubaddlem 32266 omssubadd 32267 mblfinlem2 35815 ovoliunnfl 35819 voliunnfl 35821 volsupnfl 35822 itg2addnclem 35828 radcnvrat 41932 infxrglb 42879 xrgtso 42884 infxr 42906 infxrunb2 42907 infxrpnf 42986 limsup0 43235 limsuppnfdlem 43242 limsupequzlem 43263 supcnvlimsup 43281 limsuplt2 43294 liminfval 43300 limsupge 43302 liminfgval 43303 liminfval2 43309 limsup10ex 43314 liminf10ex 43315 liminflelimsuplem 43316 cnrefiisplem 43370 etransclem48 43823 sge0val 43904 sge0z 43913 sge00 43914 sge0sn 43917 sge0tsms 43918 ovnval2 44083 smflimsuplem1 44353 smflimsuplem2 44354 smflimsuplem4 44356 smflimsuplem7 44359 |
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