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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 2735 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2735 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | ispos2 18373 | . 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
4 | 3 | simplbi 497 | 1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Proset cproset 18350 Posetcpo 18365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-proset 18352 df-poset 18371 |
This theorem is referenced by: posref 18376 isipodrs 18595 pwrssmgc 32975 mgcf1olem1 32976 mgcf1olem2 32977 mgcf1o 32978 nsgmgc 33420 ordtrest2NEWlem 33883 ordtrest2NEW 33884 ordtconnlem1 33885 |
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