Theorem posprs 18217

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 2731	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2731	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18216	. 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 2111  ∀wral 3047   class class class wbr 5086  ‘cfv 6476  Basecbs 17115  lecple 17163   Proset cproset 18193  Posetcpo 18208

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-proset 18195  df-poset 18214

This theorem is referenced by:  posref  18219  isipodrs  18438  pwrssmgc  32973  mgcf1olem1  32974  mgcf1olem2  32975  mgcf1o  32976  nsgmgc  33369  ordtrest2NEWlem  33927  ordtrest2NEW  33928  ordtconnlem1  33929  exbasprs  49008  basresprsfo  49010  discbas  49604  discthin  49605