Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10036 | . 2 ⊢ (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵) |
4 | 3 | con2bii 346 | 1 ⊢ (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-xr 9957 df-le 9959 |
This theorem is referenced by: letrii 10041 nn0ge2m1nn 11237 zgt1rpn0n1 11747 0nelfz1 12231 fzpreddisj 12260 hashnn0n0nn 13041 hashge2el2dif 13117 n2dvds1 14942 divalglem5 14958 divalglem6 14959 sadcadd 15018 strlemor1 15796 htpycc 22587 pco1 22623 pcohtpylem 22627 pcopt 22630 pcopt2 22631 pcoass 22632 pcorevlem 22634 vitalilem5 23187 vieta1lem2 23870 ppiltx 24703 ppiublem1 24727 chtub 24737 axlowdimlem16 25637 axlowdim 25641 lfgrnloop 25791 spthispth 26103 ballotlem2 29877 subfacp1lem1 30415 subfacp1lem5 30420 bcneg1 30875 poimirlem9 32588 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem22 32601 fdc 32711 pellexlem6 36416 jm2.23 36581 lfuhgr1v0e 40480 lfgrwlkprop 40896 |
Copyright terms: Public domain | W3C validator |