MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psref2 Structured version   Visualization version   GIF version

Theorem psref2 17027
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 17025 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
21ibi 255 . 2 (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅)))
32simp3d 1068 1 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031   = wceq 1475  wcel 1977  cin 3539  wss 3540   cuni 4372   I cid 4948  ccnv 5037  cres 5040  ccom 5042  Rel wrel 5043  PosetRelcps 17021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-ps 17023
This theorem is referenced by:  pslem  17029  cnvps  17035  tsrdir  17061
  Copyright terms: Public domain W3C validator