Theorem posprs 18251

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.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2737	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18250	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
4	3	simplbi 496	1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  Basecbs 17148  lecple 17196   Proset cproset 18227  Posetcpo 18242

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-proset 18229  df-poset 18248

This theorem is referenced by:  posref  18253  isipodrs  18472  pwrssmgc  33092  mgcf1olem1  33093  mgcf1olem2  33094  mgcf1o  33095  nsgmgc  33504  ordtrest2NEWlem  34099  ordtrest2NEW  34100  ordtconnlem1  34101  exbasprs  49333  basresprsfo  49335  discbas  49928  discthin  49929