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Mirrors > Home > MPE Home > Th. List > lenlti | Structured version Visualization version GIF version |
Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
lenlti | ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | lt.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | lenlt 11331 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2099 class class class wbr 5144 ℝcr 11146 < clt 11287 ≤ cle 11288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5679 df-cnv 5681 df-xr 11291 df-le 11293 |
This theorem is referenced by: ltnlei 11374 hashgt12el 14432 hashgt12el2 14433 georeclim 15869 geoisumr 15875 divalglem6 16393 umgrislfupgrlem 29053 ballotlem4 34343 signswch 34418 limsup10ex 45428 |
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