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Mirrors > Home > MPE Home > Th. List > Mathboxes > xreqnltd | Structured version Visualization version GIF version |
Description: A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
xreqnltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xreqnltd.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
xreqnltd | ⊢ (𝜑 → ¬ 𝐴 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xreqnltd.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xreqnltd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
3 | 1, 2 | eqeltrrd 2840 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
4 | xrlttri3 12877 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
6 | 1, 5 | mpbid 231 | . 2 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) |
7 | 6 | simpld 495 | 1 ⊢ (𝜑 → ¬ 𝐴 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ℝ*cxr 11008 < clt 11009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 |
This theorem is referenced by: limsuppnflem 43251 |
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