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Theorem posprs 18273
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 2732 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2732 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2ispos2 18272 . 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 498 1 (𝐾 ∈ Poset β†’ 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148  β€˜cfv 6543  Basecbs 17148  lecple 17208   Proset cproset 18250  Posetcpo 18264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-proset 18252  df-poset 18270
This theorem is referenced by:  posref  18275  isipodrs  18494  pwrssmgc  32425  mgcf1olem1  32426  mgcf1olem2  32427  mgcf1o  32428  nsgmgc  32785  ordtrest2NEWlem  33188  ordtrest2NEW  33189  ordtconnlem1  33190
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