Theorem posprs 16949

Description: A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
posprs	⊢ (𝐾 ∈ Poset → 𝐾 ∈ Preset )

Proof of Theorem posprs

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2622	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	ispos2 16948	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ Preset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
4	3	simplbi 476	1 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Preset )

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888  Basecbs 15857  lecple 15948   Preset cpreset 16926  Posetcpo 16940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-preset 16928  df-poset 16946

This theorem is referenced by:  posref  16951  isipodrs  17161  ordtrest2NEWlem  29968  ordtrest2NEW  29969  ordtconnlem1  29970