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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 2945 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
| 3 | xrlttri5d.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 4 | xrlttri5d.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 5 | xrlttri3 13185 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
| 6 | 3, 4, 5 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
| 7 | 2, 6 | mtbid 324 | . . . . 5 ⊢ (𝜑 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
| 8 | oran 992 | . . . . 5 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
| 9 | 7, 8 | sylibr 234 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
| 10 | xrlttri5d.nlt | . . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
| 11 | 9, 10 | jca 511 | . . 3 ⊢ (𝜑 → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴)) |
| 12 | pm5.61 1003 | . . 3 ⊢ (((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴) ↔ (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
| 13 | 11, 12 | sylib 218 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
| 14 | 13 | simpld 494 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ℝ*cxr 11294 < clt 11295 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: lttri5d 45311 |
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