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Mirrors > Home > MPE Home > Th. List > lelttric | Structured version Visualization version GIF version |
Description: Trichotomy law. (Contributed by NM, 4-Apr-2005.) |
Ref | Expression |
---|---|
lelttric | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 894 | . 2 ⊢ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴) | |
2 | lenlt 10708 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
3 | 2 | orbi1d 914 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴))) |
4 | 1, 3 | mpbiri 261 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∈ wcel 2111 class class class wbr 5030 ℝcr 10525 < clt 10664 ≤ cle 10665 |
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-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-xr 10668 df-le 10670 |
This theorem is referenced by: ltlecasei 10737 fzsplit2 12927 uzsplit 12974 fzospliti 13064 fzouzsplit 13067 discr1 13596 faclbnd 13646 faclbnd4lem1 13649 faclbnd4lem4 13652 dvdslelem 15651 dvdsprmpweqle 16212 icccmplem2 23428 icccmp 23430 bcmono 25861 bpos1lem 25866 bposlem3 25870 bpos 25877 fzsplit3 30543 submateq 31162 lzunuz 39709 jm2.24 39904 iccpartnel 43955 bgoldbtbnd 44327 tgoldbach 44335 reorelicc 45124 |
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