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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 11340 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2107 class class class wbr 5142 ℝcr 11155 < clt 11296 ≤ cle 11297 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-xr 11300 df-le 11302 | 
| This theorem is referenced by: ltnlei 11383 hashgt12el 14462 hashgt12el2 14463 georeclim 15909 geoisumr 15915 divalglem6 16436 umgrislfupgrlem 29140 ballotlem4 34502 signswch 34577 limsup10ex 45793 | 
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