Theorem posprs 18389

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 2765	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18388	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
4	3	simplbi 502	1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3081   class class class wbr 5111  ‘cfv 6540  Basecbs 17286  lecple 17334   Proset cproset 18365  Posetcpo 18380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-proset 18367  df-poset 18386

This theorem is used by:  posref  18391  isipodrs  18610  pwrssmgc  33343  mgcf1olem1  33344  mgcf1olem2  33345  mgcf1o  33346  nsgmgc  33744  ordtrest2NEWlem  34335  ordtrest2NEW  34336  ordtconnlem1  34337  exbasprs  49788  basresprsfo  49790  discbas  50383  discthin  50384