![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13114 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥))) | |
2 | xrlttr 13115 | . 2 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
3 | 1, 2 | isso2i 5622 | 1 ⊢ < Or ℝ* |
Colors of variables: wff setvar class |
Syntax hints: Or wor 5586 ℝ*cxr 11243 < clt 11244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 |
This theorem is referenced by: xrlttri2 13117 xrlttri3 13118 xrltne 13138 xmullem 13239 xmulasslem 13260 supxr 13288 supxrcl 13290 supxrun 13291 supxrmnf 13292 supxrunb1 13294 supxrunb2 13295 supxrub 13299 supxrlub 13300 infxrcl 13308 infxrlb 13309 infxrgelb 13310 xrinf0 13313 infmremnf 13318 limsupval 15414 limsupgval 15416 limsupgre 15421 ramval 16937 ramcl2lem 16938 prdsdsfn 17407 prdsdsval 17420 imasdsfn 17456 imasdsval 17457 prdsmet 23867 xpsdsval 23878 prdsbl 23991 tmsxpsval2 24039 nmoval 24223 xrge0tsms2 24342 metdsval 24354 iccpnfhmeo 24452 xrhmeo 24453 ovolval 24981 ovolf 24990 ovolctb 24998 itg2val 25237 mdegval 25572 mdegldg 25575 mdegxrf 25577 mdegcl 25578 aannenlem2 25833 nmooval 30003 nmoo0 30031 nmopval 31096 nmfnval 31116 nmop0 31226 nmfn0 31227 xrsupssd 31959 xrge0infssd 31961 infxrge0lb 31964 infxrge0glb 31965 infxrge0gelb 31966 xrsclat 32168 xrge0iifiso 32903 esumval 33032 esumnul 33034 esum0 33035 gsumesum 33045 esumsnf 33050 esumpcvgval 33064 esum2d 33079 omsfval 33281 omsf 33283 oms0 33284 omssubaddlem 33286 omssubadd 33287 mblfinlem2 36514 ovoliunnfl 36518 voliunnfl 36520 volsupnfl 36521 itg2addnclem 36527 radcnvrat 43058 infxrglb 44036 xrgtso 44041 infxr 44063 infxrunb2 44064 infxrpnf 44142 limsup0 44396 limsuppnfdlem 44403 limsupequzlem 44424 supcnvlimsup 44442 limsuplt2 44455 liminfval 44461 limsupge 44463 liminfgval 44464 liminfval2 44470 limsup10ex 44475 liminf10ex 44476 liminflelimsuplem 44477 cnrefiisplem 44531 etransclem48 44984 sge0val 45068 sge0z 45077 sge00 45078 sge0sn 45081 sge0tsms 45082 ovnval2 45247 smflimsuplem1 45522 smflimsuplem2 45523 smflimsuplem4 45525 smflimsuplem7 45528 |
Copyright terms: Public domain | W3C validator |