|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| 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 2737 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | tlt3.l | . . . 4 ⊢ < = (lt‘𝐾) | |
| 4 | 1, 2, 3 | tlt2 32959 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌 < 𝑋)) | 
| 5 | tospos 18465 | . . . . 5 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
| 6 | 1, 2, 3 | pleval2 18382 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) | 
| 7 | orcom 871 | . . . . . 6 ⊢ ((𝑋 < 𝑌 ∨ 𝑋 = 𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌)) | |
| 8 | 6, 7 | bitrdi 287 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌))) | 
| 9 | 5, 8 | syl3an1 1164 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌))) | 
| 10 | 9 | orbi1d 917 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∨ 𝑌 < 𝑋) ↔ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋))) | 
| 11 | 4, 10 | mpbid 232 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋)) | 
| 12 | df-3or 1088 | . 2 ⊢ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋) ↔ ((𝑋 = 𝑌 ∨ 𝑋 < 𝑌) ∨ 𝑌 < 𝑋)) | |
| 13 | 11, 12 | sylibr 234 | 1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 ltcplt 18354 Tosetctos 18461 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-proset 18340 df-poset 18359 df-plt 18375 df-toset 18462 | 
| This theorem is referenced by: archirngz 33196 archiabllem1b 33199 archiabllem2b 33203 | 
| Copyright terms: Public domain | W3C validator |