![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > posprs | Structured version Visualization version GIF version |
Description: A poset is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
posprs | ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2771 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | ispos2 17428 | . 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
4 | 3 | simplbi 490 | 1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2051 ∀wral 3081 class class class wbr 4925 ‘cfv 6185 Basecbs 16337 lecple 16426 Proset cproset 17406 Posetcpo 17420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-nul 5063 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-iota 6149 df-fv 6193 df-proset 17408 df-poset 17426 |
This theorem is referenced by: posref 17431 isipodrs 17641 ordtrest2NEWlem 30841 ordtrest2NEW 30842 ordtconnlem1 30843 |
Copyright terms: Public domain | W3C validator |