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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrleneltd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrleneltd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrleneltd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrleneltd.alb | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
xrleneltd.anb | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
xrleneltd | ⊢ (𝜑 → 𝐴 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleneltd.anb | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 1 | necomd 3068 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
3 | xrleneltd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrleneltd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrleneltd.alb | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
6 | xrleltne 12526 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | |
7 | 3, 4, 5, 6 | syl3anc 1363 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
8 | 2, 7 | mpbird 258 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 |
This theorem is referenced by: infleinf 41516 eliccelicod 41682 ge0xrre 41683 ressioosup 41707 ressiooinf 41709 sge0pr 42553 |
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