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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 2724 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2724 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | ispos2 18270 | . 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 2098 ∀wral 3053 class class class wbr 5138 ‘cfv 6533 Basecbs 17143 lecple 17203 Proset cproset 18248 Posetcpo 18262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-proset 18250 df-poset 18268 |
This theorem is referenced by: posref 18273 isipodrs 18492 pwrssmgc 32637 mgcf1olem1 32638 mgcf1olem2 32639 mgcf1o 32640 nsgmgc 32992 ordtrest2NEWlem 33391 ordtrest2NEW 33392 ordtconnlem1 33393 |
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