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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 10116 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 < clt 10074 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-xr 10078 df-le 10080 |
This theorem is referenced by: ltnlei 10158 hashgt12el 13210 hashgt12el2 13211 georeclim 14603 geoisumr 14609 divalglem6 15121 umgrislfupgrlem 26017 ballotlem4 30560 signswch 30638 |
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