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Mirrors > Home > MPE Home > Th. List > lenlt | Structured version Visualization version GIF version |
Description: 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.) |
Ref | Expression |
---|---|
lenlt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11298 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11298 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrlenlt 11317 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2an 594 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5152 ℝcr 11145 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-xr 11290 df-le 11292 |
This theorem is referenced by: ltnle 11331 letri3 11337 leloe 11338 eqlelt 11339 ne0gt0 11357 lelttric 11359 lenlti 11372 lenltd 11398 ltaddsub 11726 leord1 11779 lediv1 12117 suprleub 12218 dfinfre 12233 infregelb 12236 nnge1 12278 nnnlt1 12282 avgle1 12490 avgle2 12491 nn0nlt0 12536 recnz 12675 btwnnz 12676 prime 12681 indstr 12938 uzsupss 12962 zbtwnre 12968 rpneg 13046 2resupmax 13207 fzn 13557 nelfzo 13677 fzonlt0 13695 fllt 13811 flflp1 13812 modifeq2int 13938 om2uzlt2i 13956 fsuppmapnn0fiub0 13998 suppssfz 13999 leexp2 14175 discr 14242 bcval4 14306 ccatsymb 14572 swrd0 14648 sqrtneglem 15253 harmonic 15845 efle 16102 dvdsle 16294 dfgcd2 16529 lcmf 16611 infpnlem1 16886 pgpssslw 19576 gsummoncoe1 22234 mp2pm2mplem4 22731 dvferm1 25937 dvferm2 25939 dgrlt 26221 logleb 26557 argrege0 26565 ellogdm 26593 cxple 26649 cxple3 26655 asinneg 26838 birthdaylem3 26905 ppieq0 27128 chpeq0 27161 chteq0 27162 lgsval2lem 27260 lgsneg 27274 lgsdilem 27277 gausslemma2dlem1a 27318 gausslemma2dlem3 27321 ostth2lem1 27571 ostth3 27591 rusgrnumwwlks 29805 clwlkclwwlklem2a 29828 frgrreg 30224 friendship 30229 nmounbi 30606 nmlno0lem 30623 nmlnop0iALT 31825 supfz 35356 inffz 35357 fz0n 35358 nn0prpw 35840 leceifl 37115 poimirlem15 37141 poimirlem16 37142 poimirlem17 37143 poimirlem20 37146 poimirlem24 37150 poimirlem31 37157 poimirlem32 37158 ftc1anclem1 37199 nninfnub 37257 metakunt22 41710 ellz1 42218 rencldnfilem 42271 icccncfext 45304 subsubelfzo0 46735 digexp 47758 reorelicc 47861 |
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