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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrlttri5d | Structured version Visualization version GIF version |
Description: Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
xrlttri5d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrlttri5d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrlttri5d.aneb | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
xrlttri5d.nlt | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Ref | Expression |
---|---|
xrlttri5d | ⊢ (𝜑 → 𝐴 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlttri5d.aneb | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 1 | neneqd 3004 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | xrlttri5d.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrlttri5d.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrlttri3 12262 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
6 | 3, 4, 5 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
7 | 2, 6 | mtbid 316 | . . . . 5 ⊢ (𝜑 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
8 | oran 1017 | . . . . 5 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
9 | 7, 8 | sylibr 226 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
10 | xrlttri5d.nlt | . . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
11 | 9, 10 | jca 507 | . . 3 ⊢ (𝜑 → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴)) |
12 | pm5.61 1028 | . . 3 ⊢ (((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴) ↔ (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
13 | 11, 12 | sylib 210 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
14 | 13 | simpld 490 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4873 ℝ*cxr 10390 < clt 10391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 |
This theorem is referenced by: lttri5d 40304 |
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