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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tlt3 | Structured version Visualization version GIF version |
Description: In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
tlt3.b | ⊢ 𝐵 = (Base‘𝐾) |
tlt3.l | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
tlt3 | ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlt3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2752 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | tlt3.l | . . . 4 ⊢ < = (lt‘𝐾) | |
4 | 1, 2, 3 | tlt2 29965 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌 < 𝑋)) |
5 | tospos 29959 | . . . . 5 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
6 | 1, 2, 3 | pleval2 17158 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
7 | orcom 401 | . . . . . 6 ⊢ ((𝑋 < 𝑌 ∨ 𝑋 = 𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌)) | |
8 | 6, 7 | syl6bb 276 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌))) |
9 | 5, 8 | syl3an1 1166 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌))) |
10 | 9 | orbi1d 741 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∨ 𝑌 < 𝑋) ↔ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋))) |
11 | 4, 10 | mpbid 222 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋)) |
12 | df-3or 1073 | . 2 ⊢ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋) ↔ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋)) | |
13 | 11, 12 | sylibr 224 | 1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∨ w3o 1071 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 class class class wbr 4796 ‘cfv 6041 Basecbs 16051 lecple 16142 Posetcpo 17133 ltcplt 17134 Tosetctos 17226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-rab 3051 df-v 3334 df-sbc 3569 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-iota 6004 df-fun 6043 df-fv 6049 df-preset 17121 df-poset 17139 df-plt 17151 df-toset 17227 |
This theorem is referenced by: archirngz 30044 archiabllem1b 30047 archiabllem2b 30051 |
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