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Mirrors > Home > MPE Home > Th. List > letrii | Structured version Visualization version GIF version |
Description: Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
letrii | ⊢ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.2 | . . . 4 ⊢ 𝐵 ∈ ℝ | |
2 | lt.1 | . . . 4 ⊢ 𝐴 ∈ ℝ | |
3 | 1, 2 | ltnlei 10561 | . . 3 ⊢ (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵) |
4 | 1, 2 | ltlei 10562 | . . 3 ⊢ (𝐵 < 𝐴 → 𝐵 ≤ 𝐴) |
5 | 3, 4 | sylbir 227 | . 2 ⊢ (¬ 𝐴 ≤ 𝐵 → 𝐵 ≤ 𝐴) |
6 | 5 | orri 848 | 1 ⊢ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 833 ∈ wcel 2050 class class class wbr 4929 ℝcr 10334 < clt 10474 ≤ cle 10475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-pre-lttri 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 |
This theorem is referenced by: divalglem1 15605 |
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