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Mirrors > Home > MPE Home > Th. List > lttri3 | Structured version Visualization version GIF version |
Description: Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.) |
Ref | Expression |
---|---|
lttri3 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10324 | . 2 ⊢ < Or ℝ | |
2 | sotrieq2 5199 | . 2 ⊢ (( < Or ℝ ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
3 | 1, 2 | mpan 670 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4787 Or wor 5170 ℝcr 10141 < clt 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-pre-lttri 10216 ax-pre-lttrn 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-ltxr 10285 |
This theorem is referenced by: letri3 10329 leltne 10333 lttri3i 10358 lttri3d 10383 om2uzf1oi 12960 cos11 24500 ostth1 25543 ostth3 25548 clwlkclwwlklem2fv2 27146 clwlkclwwlklem2a 27148 frgrreggt1 27592 normgt0 28324 |
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