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Theorem posprs 17626
 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 2759 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2759 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2ispos2 17625 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
43simplbi 502 1 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112  ∀wral 3071   class class class wbr 5033  ‘cfv 6336  Basecbs 16542  lecple 16631   Proset cproset 17603  Posetcpo 17617 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-iota 6295  df-fv 6344  df-proset 17605  df-poset 17623 This theorem is referenced by:  posref  17628  isipodrs  17838  pwrssmgc  30805  mgcf1olem1  30806  mgcf1olem2  30807  mgcf1o  30808  nsgmgc  31119  ordtrest2NEWlem  31394  ordtrest2NEW  31395  ordtconnlem1  31396
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