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Theorem posprs 17429
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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2ispos2 17428 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
43simplbi 490 1 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2051  wral 3081   class class class wbr 4925  cfv 6185  Basecbs 16337  lecple 16426   Proset cproset 17406  Posetcpo 17420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-iota 6149  df-fv 6193  df-proset 17408  df-poset 17426
This theorem is referenced by:  posref  17431  isipodrs  17641  ordtrest2NEWlem  30841  ordtrest2NEW  30842  ordtconnlem1  30843
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