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Mathbox for Glauco Siliprandi |
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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 2994 | . 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 13184 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | |
7 | 3, 4, 5, 6 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
8 | 2, 7 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: infleinf 45322 pimxrneun 45439 eliccelicod 45483 ge0xrre 45484 ressioosup 45508 ressiooinf 45510 sge0pr 46350 |
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