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Theorem posprs 18269
Description: A poset is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
posprs (𝐾 ∈ Poset β†’ 𝐾 ∈ Proset )

Proof of Theorem posprs
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2733 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2ispos2 18268 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)π‘₯) β†’ π‘₯ = 𝑦)))
43simplbi 499 1 (𝐾 ∈ Poset β†’ 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3062   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204   Proset cproset 18246  Posetcpo 18260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-proset 18248  df-poset 18266
This theorem is referenced by:  posref  18271  isipodrs  18490  pwrssmgc  32170  mgcf1olem1  32171  mgcf1olem2  32172  mgcf1o  32173  nsgmgc  32523  ordtrest2NEWlem  32902  ordtrest2NEW  32903  ordtconnlem1  32904
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