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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 10406 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | mp2an 684 | 1 ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∈ wcel 2157 class class class wbr 4843 ℝcr 10223 < clt 10363 ≤ cle 10364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-xr 10367 df-le 10369 |
This theorem is referenced by: ltnlei 10448 hashgt12el 13459 hashgt12el2 13460 georeclim 14941 geoisumr 14947 divalglem6 15457 umgrislfupgrlem 26357 ballotlem4 31077 signswch 31156 |
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