Theorem posprs 18230

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 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2733	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18229	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
4	3	simplbi 497	1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  ‘cfv 6489  Basecbs 17127  lecple 17175   Proset cproset 18206  Posetcpo 18221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-proset 18208  df-poset 18227

This theorem is referenced by:  posref  18232  isipodrs  18451  pwrssmgc  33010  mgcf1olem1  33011  mgcf1olem2  33012  mgcf1o  33013  nsgmgc  33421  ordtrest2NEWlem  34007  ordtrest2NEW  34008  ordtconnlem1  34009  exbasprs  49138  basresprsfo  49140  discbas  49733  discthin  49734