Theorem posprs 18282

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 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2736	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18281	. 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 3051   class class class wbr 5085  ‘cfv 6498  Basecbs 17179  lecple 17227   Proset cproset 18258  Posetcpo 18273

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 2708  ax-nul 5241

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-proset 18260  df-poset 18279

This theorem is referenced by:  posref  18284  isipodrs  18503  pwrssmgc  33060  mgcf1olem1  33061  mgcf1olem2  33062  mgcf1o  33063  nsgmgc  33472  ordtrest2NEWlem  34066  ordtrest2NEW  34067  ordtconnlem1  34068  exbasprs  49452  basresprsfo  49454  discbas  50047  discthin  50048