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Mirrors > Home > MPE Home > Th. List > posprs | Structured version Visualization version GIF version |
Description: A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
posprs | ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Preset ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | ispos2 16771 | . 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Preset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦))) |
4 | 3 | simplbi 475 | 1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Preset ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Preset cpreset 16749 Posetcpo 16763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-preset 16751 df-poset 16769 |
This theorem is referenced by: posref 16774 isipodrs 16984 ordtrest2NEWlem 29296 ordtrest2NEW 29297 ordtconlem1 29298 |
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