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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 11407 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
| 5 | 1, 4 | mpbid 232 | 1 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-xr 11299 df-le 11301 |
| This theorem is referenced by: lbinf 12221 supaddc 12235 supmul1 12237 zsupss 12979 prodge0rd 13142 infmrp1 13386 fzdisj 13591 uzdisj 13637 fzouzdisj 13735 addmodlteq 13987 seqf1olem1 14082 seqf1olem2 14083 seqcoll 14503 seqcoll2 14504 ccatalpha 14631 rlimcld2 15614 rlimno1 15690 smupvallem 16520 lcmgcdlem 16643 4sqlem11 16993 ramcl2lem 17047 psdmul 22170 recld2 24836 nmoleub2lem3 25148 ivthlem3 25488 ovolicopnf 25559 dvferm1lem 26022 dvferm2lem 26024 dgrlb 26275 dgreq0 26305 aaliou3lem9 26392 radcnvle 26463 abelthlem2 26476 dvlog2lem 26694 lgsval2lem 27351 pntlem3 27653 irredminply 33757 unblimceq0lem 36507 unblimceq0 36508 mblfinlem2 37665 imo72b2 44185 climisp 45761 stoweidlem52 46067 fourierdlem10 46132 fourierdlem12 46134 fourierdlem20 46142 fourierdlem50 46171 fourierdlem54 46175 fourierdlem103 46224 fouriersw 46246 etransclem35 46284 etransc 46298 |
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