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Mirrors > Home > MPE Home > Th. List > ltso | Structured version Visualization version GIF version |
Description: 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
Ref | Expression |
---|---|
ltso | ⊢ < Or ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlttri 10448 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥))) | |
2 | lttr 10453 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
3 | 1, 2 | isso2i 5308 | 1 ⊢ < Or ℝ |
Colors of variables: wff setvar class |
Syntax hints: Or wor 5273 ℝcr 10271 < clt 10411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 |
This theorem is referenced by: gtso 10458 lttri2 10459 lttri3 10460 lttri4 10461 ltnr 10471 ltnsym2 10475 fimaxre 11322 lbinf 11330 suprcl 11337 suprub 11338 suprlub 11341 infrecl 11359 infregelb 11361 infrelb 11362 supfirege 11363 suprfinzcl 11844 uzinfi 12075 suprzcl2 12085 suprzub 12086 2resupmax 12331 infmrp1 12486 fseqsupcl 13095 ssnn0fi 13103 fsuppmapnn0fiublem 13108 isercolllem1 14803 isercolllem2 14804 summolem2 14854 zsum 14856 fsumcvg3 14867 mertenslem2 15020 prodmolem2 15068 zprod 15070 cnso 15380 gcdval 15624 dfgcd2 15669 lcmval 15711 lcmgcdlem 15725 odzval 15900 pczpre 15956 prmreclem1 16024 ramz 16133 odval 18337 odf 18340 gexval 18377 gsumval3 18694 retos 20361 mbfsup 23868 mbfinf 23869 itg2monolem1 23954 itg2mono 23957 dvgt0lem2 24203 dvgt0 24204 plyeq0lem 24403 dgrval 24421 dgrcl 24426 dgrub 24427 dgrlb 24429 elqaalem1 24511 elqaalem3 24513 aalioulem2 24525 logccv 24846 ex-po 27867 ssnnssfz 30113 lmdvg 30597 oddpwdc 31014 ballotlemi 31161 ballotlemiex 31162 ballotlemsup 31165 ballotlemimin 31166 ballotlemfrcn0 31190 ballotlemirc 31192 erdszelem3 31774 erdszelem4 31775 erdszelem5 31776 erdszelem6 31777 erdszelem8 31779 erdszelem9 31780 erdszelem11 31782 erdsze2lem1 31784 erdsze2lem2 31785 supfz 32208 inffz 32209 gtinf 32902 ptrecube 34030 poimirlem31 34061 poimirlem32 34062 heicant 34065 mblfinlem3 34069 mblfinlem4 34070 ismblfin 34071 incsequz2 34164 totbndbnd 34207 prdsbnd 34211 pellfundval 38397 dgraaval 38666 dgraaf 38669 fzisoeu 40416 uzublem 40556 infrglb 40723 limsupubuzlem 40845 fourierdlem25 41269 fourierdlem31 41275 fourierdlem36 41280 fourierdlem37 41281 fourierdlem42 41286 fourierdlem79 41322 ioorrnopnlem 41441 hoicvr 41682 hoidmvlelem2 41730 iunhoiioolem 41809 vonioolem1 41814 prmdvdsfmtnof1lem1 42510 prmdvdsfmtnof 42512 prmdvdsfmtnof1 42513 ssnn0ssfz 43135 rrx2plordso 43453 |
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