![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrltnled | Structured version Visualization version GIF version |
Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
xrltnled.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrltnled.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
xrltnled | ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnled.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrltnled.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xrltnle 11268 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2107 class class class wbr 5144 ℝ*cxr 11234 < clt 11235 ≤ cle 11236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-cnv 5680 df-le 11241 |
This theorem is referenced by: infxrbnd2 43952 infleinflem2 43954 xrralrecnnge 43973 qinioo 44121 limsuppnflem 44299 limsupre2lem 44313 meaiuninc3v 45073 ovolval4lem1 45238 preimagelt 45288 preimalegt 45289 |
Copyright terms: Public domain | W3C validator |