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| Mirrors > Home > MPE Home > Th. List > xrltnle | Structured version Visualization version GIF version | ||
| Description: "Less than" expressed in terms of "less than or equal to", for extended reals. (Contributed by NM, 6-Feb-2007.) |
| Ref | Expression |
|---|---|
| xrltnle | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrlenlt 11326 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
| 2 | 1 | con2bid 354 | . 2 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
| 3 | 2 | ancoms 458 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-le 11301 |
| This theorem is referenced by: xrletri 13195 qextltlem 13244 xralrple 13247 xltadd1 13298 xsubge0 13303 xposdif 13304 xltmul1 13334 ioo0 13412 ico0 13433 ioc0 13434 snunioo 13518 snunioc 13520 difreicc 13524 hashbnd 14375 limsuplt 15515 pcadd 16927 pcadd2 16928 ramubcl 17056 ramlb 17057 leordtvallem1 23218 leordtvallem2 23219 leordtval2 23220 leordtval 23221 lecldbas 23227 blcld 24518 stdbdbl 24530 tmsxpsval2 24552 iocmnfcld 24789 xrsxmet 24831 metdsge 24871 bndth 24990 ovolgelb 25515 ovolunnul 25535 ioombl 25600 volsup2 25640 mbfmax 25684 ismbf3d 25689 itg2seq 25777 itg2monolem2 25786 itg2monolem3 25787 lhop2 26054 mdegleb 26103 deg1ge 26137 deg1add 26142 ig1pdvds 26219 plypf1 26251 radcnvlt1 26461 upgrfi 29108 xrdifh 32782 xrge00 33017 gsumesum 34060 itg2gt0cn 37682 asindmre 37710 dvasin 37711 aks6d1c6lem3 42173 aks6d1c7lem2 42182 iocioodisjd 42355 radcnvrat 44333 supxrgelem 45348 infrpge 45362 xrlexaddrp 45363 xrltnled 45374 xrpnf 45496 gtnelioc 45504 ltnelicc 45510 gtnelicc 45513 snunioo1 45525 eliccnelico 45542 xrgtnelicc 45551 lptioo2 45646 stoweidlem34 46049 fourierdlem20 46142 fouriersw 46246 nltle2tri 47325 iccelpart 47420 |
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