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Mirrors > Home > MPE Home > Th. List > ltnlei | Structured version Visualization version GIF version |
Description: 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
ltnlei | ⊢ (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
2 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
3 | 1, 2 | lenlti 11087 | . 2 ⊢ (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵) |
4 | 3 | con2bii 358 | 1 ⊢ (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2110 class class class wbr 5079 ℝcr 10863 < clt 11002 ≤ cle 11003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5595 df-cnv 5597 df-xr 11006 df-le 11008 |
This theorem is referenced by: letrii 11092 nn0ge2m1nn 12294 0nelfz1 13266 fzpreddisj 13296 hashnn0n0nn 14096 hashge2el2dif 14184 divalglem5 16096 divalglem6 16097 sadcadd 16155 htpycc 24133 pco1 24168 pcohtpylem 24172 pcopt 24175 pcopt2 24176 pcoass 24177 pcorevlem 24179 vitalilem5 24766 vieta1lem2 25461 ppiltx 26316 ppiublem1 26340 chtub 26350 axlowdimlem16 27315 axlowdim 27319 lfgrnloop 27485 lfuhgr1v0e 27611 lfgrwlkprop 28044 ballotlem2 32443 subfacp1lem1 33129 subfacp1lem5 33134 bcneg1 33690 poimirlem9 35774 poimirlem16 35781 poimirlem17 35782 poimirlem19 35784 poimirlem20 35785 poimirlem22 35787 fdc 35891 pellexlem6 40645 jm2.23 40807 |
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