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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 10702 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5058 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-le 10675 |
This theorem is referenced by: infxrbnd2 41630 infleinflem2 41632 xrralrecnnge 41655 qinioo 41804 limsuppnflem 41984 limsupre2lem 41998 meaiuninc3v 42760 ovolval4lem1 42925 preimagelt 42974 preimalegt 42975 |
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