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Theorem posprs 17547
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 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2818 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2ispos2 17546 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
43simplbi 498 1 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560   Proset cproset 17524  Posetcpo 17538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-proset 17526  df-poset 17544
This theorem is referenced by:  posref  17549  isipodrs  17759  ordtrest2NEWlem  31064  ordtrest2NEW  31065  ordtconnlem1  31066
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