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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tleile | Structured version Visualization version GIF version |
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
tleile.b | ⊢ 𝐵 = (Base‘𝐾) |
tleile.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
tleile | ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1128 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
2 | simp3 1129 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
3 | tleile.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
4 | tleile.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | istos 17425 | . . . 4 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))) |
6 | 5 | simprbi 492 | . . 3 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
7 | 6 | 3ad2ant1 1124 | . 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) |
8 | breq1 4891 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦)) | |
9 | breq2 4892 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
10 | 8, 9 | orbi12d 905 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋))) |
11 | breq2 4892 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌)) | |
12 | breq1 4891 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋)) | |
13 | 11, 12 | orbi12d 905 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))) |
14 | 10, 13 | rspc2va 3525 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)) |
15 | 1, 2, 7, 14 | syl21anc 828 | 1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 class class class wbr 4888 ‘cfv 6137 Basecbs 16259 lecple 16349 Posetcpo 17330 Tosetctos 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-iota 6101 df-fv 6145 df-toset 17424 |
This theorem is referenced by: tltnle 30228 odutos 30229 trleile 30232 |
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