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| Mirrors > Home > MPE Home > Th. List > ltnri | Structured version Visualization version GIF version | ||
| Description: 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
| Ref | Expression |
|---|---|
| lt.1 | ⊢ 𝐴 ∈ ℝ |
| Ref | Expression |
|---|---|
| ltnri | ⊢ ¬ 𝐴 < 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lt.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
| 2 | ltnr 11356 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 < 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 class class class wbr 5143 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: lt0ne0d 11828 prodgt0 12114 elnnnn0b 12570 0nrp 13070 geolim 15906 geolim2 15907 georeclim 15908 geoisum1c 15916 0ringnnzr 20525 dscopn 24586 logcnlem3 26686 jensen 27032 gausslemma2dlem0i 27408 2sqreultblem 27492 2sqreunnltblem 27495 ostth 27683 tgcgr4 28539 clwwlkn0 30047 konigsberg 30276 expgt0b 32818 fldext2chn 33769 signswch 34576 signlem0 34602 poimirlem32 37659 oexpreposd 42357 pell1qrgaplem 42884 relexp01min 43726 rexanuz2nf 45503 sbgoldbaltlem1 47766 ex-gt 49247 |
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