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Theorem posprs 18374
Description: A poset is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
posprs (𝐾 ∈ Poset → 𝐾 ∈ Proset )

Proof of Theorem posprs
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2ispos2 18373 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
43simplbi 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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