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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrlelttric | Structured version Visualization version GIF version |
Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
Ref | Expression |
---|---|
xrlelttric | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 432 | . 2 ⊢ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴) | |
2 | xrlenlt 9982 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
3 | 2 | orbi1d 735 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴))) |
4 | 1, 3 | mpbiri 247 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-le 9959 |
This theorem is referenced by: difioo 28934 esumpcvgval 29467 |
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