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Theorem psref2 17185
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 17183 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 256 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp3d 1073 1 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  cin 3566  wss 3567   cuni 4427   I cid 5013  ccnv 5103  cres 5106  ccom 5108  Rel wrel 5109  PosetRelcps 17179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-res 5116  df-ps 17181
This theorem is referenced by:  pslem  17187  cnvps  17193  tsrdir  17219
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