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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 10711 | . 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 2113 class class class wbr 5069 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-cnv 5566 df-le 10684 |
This theorem is referenced by: infxrbnd2 41643 infleinflem2 41645 xrralrecnnge 41668 qinioo 41817 limsuppnflem 41997 limsupre2lem 42011 meaiuninc3v 42773 ovolval4lem1 42938 preimagelt 42987 preimalegt 42988 |
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