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Theorem psref2 17930
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 17928 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 270 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp3d 1145 1 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  cin 3842  wss 3843   cuni 4796   I cid 5428  ccnv 5524  cres 5527  ccom 5529  Rel wrel 5530  PosetRelcps 17924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-in 3850  df-ss 3860  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-res 5537  df-ps 17926
This theorem is referenced by:  pslem  17932  cnvps  17938  tsrdir  17964
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