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Mirrors > Home > MPE Home > Th. List > lensymd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lensymd.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
lensymd | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lensymd.3 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | lenltd 11356 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbid 231 | 1 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 class class class wbr 5147 ℝcr 11105 < clt 11244 ≤ cle 11245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-xr 11248 df-le 11250 |
This theorem is referenced by: lbinf 12163 supaddc 12177 supmul1 12179 zsupss 12917 prodge0rd 13077 infmrp1 13319 fzdisj 13524 uzdisj 13570 fzouzdisj 13664 addmodlteq 13907 seqf1olem1 14003 seqf1olem2 14004 seqcoll 14421 seqcoll2 14422 ccatalpha 14539 rlimcld2 15518 rlimno1 15596 smupvallem 16420 lcmgcdlem 16539 4sqlem11 16884 ramcl2lem 16938 recld2 24321 nmoleub2lem3 24622 ivthlem3 24961 ovolicopnf 25032 dvferm1lem 25492 dvferm2lem 25494 dgrlb 25741 dgreq0 25770 aaliou3lem9 25854 radcnvle 25923 abelthlem2 25935 dvlog2lem 26151 lgsval2lem 26799 pntlem3 27101 unblimceq0lem 35370 unblimceq0 35371 mblfinlem2 36514 imo72b2 42909 climisp 44448 stoweidlem52 44754 fourierdlem10 44819 fourierdlem12 44821 fourierdlem20 44829 fourierdlem50 44858 fourierdlem54 44862 fourierdlem103 44911 fouriersw 44933 etransclem35 44971 etransc 44985 |
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