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Mirrors > Home > MPE Home > Th. List > nltled | Structured version Visualization version GIF version |
Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
nltled.1 | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Ref | Expression |
---|---|
nltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltled.1 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | lenltd 10786 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 < clt 10675 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-xr 10679 df-le 10681 |
This theorem is referenced by: dedekind 10803 suprub 11602 infrelb 11626 suprzub 12340 prodge0rd 12497 seqf1olem1 13410 bitsfzolem 15783 bitsmod 15785 reconnlem2 23435 ioombl1lem4 24162 dgrub 24824 dgrlb 24826 1smat1 31069 imo72b2 40545 dvbdfbdioolem2 42234 stoweidlem14 42319 fourierdlem10 42422 fourierdlem12 42424 fourierdlem20 42432 fourierdlem24 42436 fourierdlem50 42461 fourierdlem54 42465 fourierdlem63 42474 fourierdlem65 42476 fourierdlem75 42486 fourierdlem79 42490 fouriersw 42536 etransclem3 42542 etransclem7 42546 etransclem10 42549 etransclem15 42554 etransclem20 42559 etransclem21 42560 etransclem22 42561 etransclem24 42563 etransclem25 42564 etransclem27 42566 etransclem32 42571 |
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