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Theorem psref2 18628
Description: A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
psref2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))

Proof of Theorem psref2
StepHypRef Expression
1 isps 18626 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 267 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp3d 1143 1 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963   cuni 4912   I cid 5582  ccnv 5688  cres 5691  ccom 5693  Rel wrel 5694  PosetRelcps 18622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-res 5701  df-ps 18624
This theorem is referenced by:  pslem  18630  cnvps  18636  tsrdir  18662
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