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Theorem posprs 16881
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.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2ispos2 16880 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ Preset ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
43simplbi 476 1 (𝐾 ∈ Poset → 𝐾 ∈ Preset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907   class class class wbr 4618  cfv 5852  Basecbs 15792  lecple 15880   Preset cpreset 16858  Posetcpo 16872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-preset 16860  df-poset 16878
This theorem is referenced by:  posref  16883  isipodrs  17093  ordtrest2NEWlem  29774  ordtrest2NEW  29775  ordtconnlem1  29776
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