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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 2935 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
3 | xrlttri5d.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrlttri5d.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrlttri3 13154 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
6 | 3, 4, 5 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
7 | 2, 6 | mtbid 323 | . . . . 5 ⊢ (𝜑 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
8 | oran 987 | . . . . 5 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | |
9 | 7, 8 | sylibr 233 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) |
10 | xrlttri5d.nlt | . . . 4 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
11 | 9, 10 | jca 510 | . . 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 217 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
14 | 13 | simpld 493 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 ℝ*cxr 11277 < clt 11278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 |
This theorem is referenced by: lttri5d 44744 |
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