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Mathbox for Glauco Siliprandi |
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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 11325 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2105 class class class wbr 5147 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-le 11298 |
This theorem is referenced by: infxrbnd2 45318 infleinflem2 45320 xrralrecnnge 45339 qinioo 45487 limsuppnflem 45665 limsupre2lem 45679 meaiuninc3v 46439 ovolval4lem1 46604 preimagelt 46654 preimalegt 46655 |
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