Theorem posprs 18483

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 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18482	. 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 2145  ∀wral 3077   class class class wbr 5103  ‘cfv 6537  Basecbs 17380  lecple 17428   Proset cproset 18459  Posetcpo 18474

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 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-proset 18461  df-poset 18480

This theorem is used by:  posref  18485  isipodrs  18704  pwrssmgc  33554  mgcf1olem1  33555  mgcf1olem2  33556  mgcf1o  33557  nsgmgc  33956  ordtrest2NEWlem  34547  ordtrest2NEW  34548  ordtconnlem1  34549  exbasprs  50054  basresprsfo  50056  discbas  50649  discthin  50650