Theorem posprs 18324

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 2756	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2756	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 18323	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Proset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
4	3	simplbi 499	1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2136  ∀wral 3070   class class class wbr 5094  ‘cfv 6510  Basecbs 17221  lecple 17269   Proset cproset 18300  Posetcpo 18315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-nul 5250

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-iota 6466  df-fv 6518  df-proset 18302  df-poset 18321

This theorem is referenced by:  posref  18326  isipodrs  18545  pwrssmgc  33132  mgcf1olem1  33133  mgcf1olem2  33134  mgcf1o  33135  nsgmgc  33552  ordtrest2NEWlem  34173  ordtrest2NEW  34174  ordtconnlem1  34175  exbasprs  49546  basresprsfo  49548  discbas  50141  discthin  50142