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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 2992 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | xrlttri5d.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrlttri5d.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrlttri3 12524 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
6 | 3, 4, 5 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
7 | 2, 6 | mtbid 327 | . . . . 5 ⊢ (𝜑 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
8 | oran 987 | . . . . 5 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
9 | 7, 8 | sylibr 237 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
10 | xrlttri5d.nlt | . . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
11 | 9, 10 | jca 515 | . . 3 ⊢ (𝜑 → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴)) |
12 | pm5.61 998 | . . 3 ⊢ (((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ∧ ¬ 𝐵 < 𝐴) ↔ (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
13 | 11, 12 | sylib 221 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
14 | 13 | simpld 498 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ℝ*cxr 10663 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: lttri5d 41931 |
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