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| Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18383. (Contributed by Thierry Arnoux, 11-Feb-2018.) | 
| Ref | Expression | 
|---|---|
| tleile.b | ⊢ 𝐵 = (Base‘𝐾) | 
| tleile.l | ⊢ ≤ = (le‘𝐾) | 
| tltnle.s | ⊢ < = (lt‘𝐾) | 
| Ref | Expression | 
|---|---|
| tltnle | ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tospos 18465 | . . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
| 2 | tleile.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 3 | tleile.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 4 | tltnle.s | . . . 4 ⊢ < = (lt‘𝐾) | |
| 5 | 2, 3, 4 | pltval3 18384 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))) | 
| 6 | 1, 5 | syl3an1 1164 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))) | 
| 7 | 2, 3 | tleile 18466 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)) | 
| 8 | ibar 528 | . . . 4 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → (¬ 𝑌 ≤ 𝑋 ↔ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋))) | |
| 9 | pm5.61 1003 | . . . 4 ⊢ (((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)) | |
| 10 | 8, 9 | bitr2di 288 | . . 3 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋)) | 
| 11 | 7, 10 | syl 17 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋)) | 
| 12 | 6, 11 | bitrd 279 | 1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ 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-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: tlt2 32959 toslublem 32962 tosglblem 32964 isarchi2 33192 archirng 33195 archiabllem2c 33202 archiabl 33205 | 
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