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 2622 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | tlt3.l | . . . 4 ⊢ < = (lt‘𝐾) | |
4 | 1, 2, 3 | tlt2 29664 | . . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌 < 𝑋)) |
5 | tospos 29658 | . . . . 5 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset) | |
6 | 1, 2, 3 | pleval2 16965 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
7 | orcom 402 | . . . . . 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 1359 | . . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌 ∨ 𝑋 < 𝑌))) |
10 | 9 | orbi1d 739 | . . 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 1038 | . 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 383 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 Posetcpo 16940 ltcplt 16941 Tosetctos 17033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 |
This theorem is referenced by: archirngz 29743 archiabllem1b 29746 archiabllem2b 29750 |
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