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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 11332 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 < 𝑥))) | |
| 2 | lttr 11337 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 < 𝑧) → 𝑥 < 𝑧)) | |
| 3 | 1, 2 | isso2i 5629 | 1 ⊢ < Or ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: Or wor 5591 ℝcr 11154 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: gtso 11342 lttri2 11343 lttri3 11344 lttri4 11345 ltnr 11356 ltnsym2 11360 fimaxre 12212 fiminre 12215 lbinf 12221 suprcl 12228 suprub 12229 suprlub 12232 infrecl 12250 infregelb 12252 infrelb 12253 supfirege 12255 suprfinzcl 12732 uzinfi 12970 suprzcl2 12980 suprzub 12981 2resupmax 13230 infmrp1 13386 fseqsupcl 14018 ssnn0fi 14026 fsuppmapnn0fiublem 14031 isercolllem1 15701 isercolllem2 15702 summolem2 15752 zsum 15754 fsumcvg3 15765 mertenslem2 15921 prodmolem2 15971 zprod 15973 cnso 16283 gcdval 16533 dfgcd2 16583 lcmval 16629 lcmgcdlem 16643 odzval 16829 pczpre 16885 prmreclem1 16954 ramz 17063 odval 19552 odf 19555 gexval 19596 gsumval3 19925 retos 21636 mbfsup 25699 mbfinf 25700 itg2monolem1 25785 itg2mono 25788 dvgt0lem2 26042 dvgt0 26043 plyeq0lem 26249 dgrval 26267 dgrcl 26272 dgrub 26273 dgrlb 26275 elqaalem1 26361 elqaalem3 26363 aalioulem2 26375 logccv 26705 ex-po 30454 ssnnssfz 32789 lmdvg 33952 oddpwdc 34356 ballotlemi 34503 ballotlemiex 34504 ballotlemsup 34507 ballotlemimin 34508 ballotlemfrcn0 34532 ballotlemirc 34534 erdszelem3 35198 erdszelem4 35199 erdszelem5 35200 erdszelem6 35201 erdszelem8 35203 erdszelem9 35204 erdszelem11 35206 erdsze2lem1 35208 erdsze2lem2 35209 supfz 35729 inffz 35730 gtinf 36320 ptrecube 37627 poimirlem31 37658 poimirlem32 37659 heicant 37662 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 incsequz2 37756 totbndbnd 37796 prdsbnd 37800 aks4d1p4 42080 aks4d1p7 42084 sticksstones1 42147 sticksstones3 42149 sn-suprcld 42503 sn-suprubd 42504 pellfundval 42891 dgraaval 43156 dgraaf 43159 fzisoeu 45312 uzublem 45441 infrglb 45605 limsupubuzlem 45727 fourierdlem25 46147 fourierdlem31 46153 fourierdlem36 46158 fourierdlem37 46159 fourierdlem42 46164 fourierdlem79 46200 ioorrnopnlem 46319 hoicvr 46563 hoidmvlelem2 46611 iunhoiioolem 46690 vonioolem1 46695 fsupdm2 46858 finfdm2 46862 prmdvdsfmtnof1lem1 47571 prmdvdsfmtnof 47573 prmdvdsfmtnof1 47574 ssnn0ssfz 48265 rrx2plordso 48645 |
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